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In this video, weโll learn how to use the properties of isosceles triangles to find a missing angle or side length.
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Weโll also look at how we can do this when the given angles or sides are as algebraic expressions.
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Letโs begin with the definition of an isosceles triangle.
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An isosceles triangle is defined as a triangle which has two of its sides equal in length.
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So in this isosceles triangle ๐ด๐ต๐ถ, we can see from the hash marks that ๐ด๐ต is equal in length to ๐ด๐ถ.
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The unequal side is usually called the base of the isosceles triangle, and these would be the base angles.
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In this case, it would be angle ๐ต and angle ๐ถ.
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This leads us to another important property of an isosceles triangle, and that is that the base angles are always equal.
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You probably already know that isosceles triangles have two equal angles.
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But have you ever thought about why?
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Letโs take our isosceles triangle ๐ด๐ต๐ถ and see if we can prove that there are two equal angles.
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We start by drawing a median ๐ด๐ฆ.
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This is a line from ๐ด going to the midpoint of line segment ๐ต๐ถ.
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Because ๐ฆ is a midpoint, we can say that the line segment ๐ต๐ฆ is equal to the line segment ๐ฆ๐ถ.
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Next, because we have an isosceles triangle, we know that thereโs two equal sides.
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In this case, we were given that line segment ๐ด๐ต is equal to line segment ๐ด๐ถ.
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Finally, as we consider the two smaller triangles, triangle ๐ด๐ต๐ฆ and triangle ๐ด๐ถ๐ฆ, we know that they have a common side, which is the line segment ๐ด๐ฆ.
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So as we consider our two triangles, ๐ด๐ต๐ฆ on the left and triangle ๐ด๐ถ๐ฆ on the right, we can in fact say that these two triangles are congruent.
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And thatโs because weโve demonstrated that there are three pairs of sides congruent.
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So we can use the SSS congruency criterion.
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Importantly for us, it means that this angle at ๐ต is equal to the angle at ๐ถ, and so demonstrating the property that these angles are always equal in an isosceles triangle.
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As we go through our questions on isosceles triangles, weโll be using the fact that they have two equal sides and two equal angles.
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So letโs have a look at our first question.
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Find the measure of angle ๐ด๐ต๐ถ.
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In the diagram, we have a triangle.
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Weโre given the angle ๐ถ๐ด๐ต as 58 degrees.
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And we need to work out this angle of ๐ด๐ต๐ถ.
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We should notice that on this diagram, we have two hash markings on the lines, indicating that thereโs two equal sides.
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So we could say that triangle ๐ด๐ต๐ถ is in fact an isosceles triangle, as an isosceles triangle is defined as a triangle that has two sides equal in length.
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An important property of isosceles triangles is that base angles are equal.
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In an isosceles triangle, the unequal side is referred to as the base, so that would be this line ๐ด๐ต.
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And the base angles would be the two angles adjacent to this base.
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But be careful because the base isnโt always at the bottom of the diagram.
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We can therefore write that angle ๐ด๐ต๐ถ is equal to angle ๐ต๐ด๐ถ.
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As weโre given that angle ๐ต๐ด๐ถ is 58 degrees, then so is angle ๐ด๐ต๐ถ.
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We can give our answer then that the measure of angle ๐ด๐ต๐ถ is 58 degrees.
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Letโs have a look at another question involving the angles in an isosceles triangle.
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In the displayed model of a house, what angle does the roof make with the horizontal, given that triangle ๐ด๐ธ๐ต is isosceles?
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Itโs probably most useful to begin this question by highlighting the triangle ๐ด๐ธ๐ต that we want to consider.
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This triangle forms the roof of this house, and weโre told that itโs isosceles.
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We should recall that in an isosceles triangle, we have two sides equal in length and two base angles are equal.
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So in this diagram, the side ๐ด๐ธ is equal to the side ๐ด๐ต and the angle ๐ด๐ธ๐ต is equal to the angle ๐ด๐ต๐ธ.
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Now that weโve had a look at the diagram, letโs focus on what weโre asked, to find the angle that the roof makes with the horizontal.
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That means that weโre really looking for the angle created by the slope of the roof and the horizontal axis.
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Either of the two base angles would give us the answer for this.
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So letโs see if we can work out one of these angles, angle ๐ด๐ต๐ธ.
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In order to do this, weโll need to remember an important fact about the angles in a triangle.
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And that is that the angles in a triangle add up to 180 degrees.
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This means that we can write that angle ๐ต๐ด๐ธ plus angle ๐ด๐ธ๐ต plus angle ๐ด๐ต๐ธ is equal to 180 degrees.
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Weโre given that angle ๐ต๐ด๐ธ is 107 degrees.
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So if we subtract 107 degrees from both sides of this equation, we get that angle ๐ด๐ธ๐ต plus angle ๐ด๐ต๐ธ is equal to 73 degrees.
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As we have an isosceles triangle, we know that our two base angles are equal.
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So angle ๐ด๐ธ๐ต is equal to angle ๐ด๐ต๐ธ.
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We could think of this then that two times angle ๐ด๐ต๐ธ is 73 degrees.
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And so to find angle ๐ด๐ต๐ธ, we must divide both sides of this equation by two, which means that the measure of angle ๐ด๐ต๐ธ is 36.5 degrees as a decimal.
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We can therefore give our answer that the angle that the roof makes with the horizontal is 36.5 degrees.
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This next question looks a little more complicated.
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But donโt worry.
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Weโll just need to apply some of our algebra skills along with the properties that we know of isosceles triangles.
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Find the values of ๐ฅ and ๐ฆ.
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In this triangle question, the important thing to notice is these markings on the line, which indicate that itโs an isosceles triangle, as we have two sides of equal length.
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We should also remember that isosceles triangles have two equal angles.
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In this case, itโs these two base angles of nine ๐ฆ minus three degrees and ๐ฅ plus one degrees.
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We might begin trying to solve this by writing the equation that nine ๐ฆ minus three degrees equals ๐ฅ plus one degrees.
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However, in order to solve an equation like this that has two unknowns, the ๐ฅ and ๐ฆ, weโd need another equation linking ๐ฅ and ๐ฆ, which we donโt have.
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So letโs try another approach.
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This time, weโll try thinking about the sum of the angles in a triangle.
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Using the fact that the angles in a triangle add up to 180 degrees might give us a way to find out an actual numerical value for nine ๐ฆ minus three and ๐ฅ plus one.
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If we write that 96 degrees plus nine ๐ฆ minus three degrees plus ๐ฅ plus one degrees equals 180 degrees, then we could subtract 96 degrees from both sides of this equation to give us that these two values of nine ๐ฆ minus three degrees plus ๐ฅ plus one degrees must equal 84 degrees.
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And since we know that these two angles are equal to each other, that means that each of these must be equal to half of 84 degrees.
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So nine ๐ฆ minus three degrees must be 42 degrees, and ๐ฅ plus one degrees must be 42 degrees.
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Writing these two equations and solving them would give us the values of ๐ฅ and ๐ฆ.
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Beginning with nine ๐ฆ minus three degrees equals 42 degrees, we could add three to both sides.
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So nine ๐ฆ is equal to 45.
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Dividing both sides by nine would give us that ๐ฆ equals five.
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In order to solve the second equation, ๐ฅ plus one degrees equals 42 degrees, we just need to subtract one from both sides.
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So ๐ฅ is equal to 41.
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Before we finish, itโs always worth checking our values.
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If ๐ฅ is 41, then this angle is 42 degrees.
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If ๐ฆ is five, then this angle is also 42 degrees.
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The first check could be that these two angles are indeed equal.
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And the final check would be to see if 42 plus 42 plus 96 does indeed give 180 degrees, and it does.
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So we can give our answers that ๐ฅ is equal to 41 and ๐ฆ is equal to five.
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Letโs have a look at another question.
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Which of the following is true?
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Option (A) ๐ด๐ต equals ๐ด๐ถ, option (B) ๐ถ๐ด equals ๐ถ๐ต, option (C) ๐ต๐ถ equals ๐ต๐ด.
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In the diagram, we have a triangle ๐ด๐ต๐ถ, weโve got a line ๐ต๐ถ, and weโve got this ray ๐ต๐ด.
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In the options that weโre given, weโre really looking to see if thereโs a pair of lines which are equal in length.
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Weโre not given any hash marks on any of the lines, which would indicate that two are equal.
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So letโs have a look at the angles instead.
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If we use the fact that the angles on a straight line add up to 180 degrees, then we should be able to work out this angle ๐ด๐ถ๐ต and this angle ๐ด๐ต๐ถ.
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Starting with angle ๐ด๐ถ๐ต, we can write that thatโs equal to 180 degrees subtract 98 degrees.
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We can work this out by calculating 180 degrees subtract 100 degrees and then adding on two, which would give us a value of 82 degrees.
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We can add this value to the diagram.
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The next angle, angle ๐ด๐ต๐ถ, must be equal to 180 degrees subtract 131 degrees.
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Subtracting 130 degrees and then another one degree would give us 49 degrees.
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Now that weโve found these two angles in this diagram, we might think that itโs not very helpful.
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But letโs see if we can have a look at calculating the other angle in the triangle ๐ด๐ต๐ถ.
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We should remember that the angles in a triangle add up to 180 degrees.
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So weโll have our unknown angle ๐ถ๐ด๐ต plus angle ๐ด๐ถ๐ต, which we worked out as 82 degrees, plus the angle ๐ด๐ต๐ถ, which we worked out as 49 degrees, all must add to give 180 degrees.
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This means angle ๐ถ๐ด๐ต plus 131 degrees equals 180 degrees.
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Subtracting 131 degrees from both sides of this equation gives us that angle ๐ถ๐ด๐ต is equal to 49 degrees.
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We can then add that onto our diagram and see if thereโs anything to notice.
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Well, we should hopefully see that we have in fact got two angles that are the same size.
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Both of these are 49 degrees.
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This means that weโve got an isosceles triangle here in triangle ๐ด๐ต๐ถ.
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Any triangle that has two equal angles must have two equal sides.
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And therefore, itโs an isosceles triangle.
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The two sides that are equal will be this side ๐ถ๐ด and this side ๐ถ๐ต.
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We can then give our answer that ๐ถ๐ด equals ๐ถ๐ต, which was the option given in option (B).
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In the final question, weโll find the area of an isosceles triangle.
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Calculate the area of the triangle ๐ด๐ต๐ถ.
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The first thing we might note when weโre looking at this triangle is that itโs an isosceles triangle.
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We can tell this from the hash notation indicating that we have two sides of equal length, which therefore fits the definition of an isosceles triangle.
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In order to find the area of a triangle, weโll need the formula that the area of a triangle is equal to half multiplied by the base multiplied by the perpendicular height.
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If we were to try and immediately calculate this however, weโd have a problem.
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This value of 10 centimeters represents the slant height of the triangle but not the perpendicular height.
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The perpendicular height would look like this.
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We can even define it with the letter โ if we wish.
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As weโve created two right triangles here, we might consider the Pythagorean theorem, which tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides.
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So letโs consider this triangle on the left.
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We can say that this line from ๐ด meets the line ๐ต๐ถ at point ๐ฆ.
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In order to use the Pythagorean theorem, we need to know the length of this line segment ๐ต๐ฆ.
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Now, you might think that itโs very clear that itโs six centimeters.
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But how can we be absolutely sure that it is six centimeters?
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Letโs consider the two triangles.
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Weโve got triangle ๐ด๐ต๐ฆ on the left and triangle ๐ด๐ถ๐ฆ on the right.
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If we consider a pair of sides, we know that side ๐ด๐ต is equal to side ๐ด๐ถ.
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Side ๐ด๐ฆ is common to both triangles.
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And finally, angle ๐ด๐ฆ๐ต is equal to angle ๐ด๐ฆ๐ถ.
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Theyโre both 90 degrees.
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We can say then that triangle ๐ด๐ต๐ฆ is congruent with triangle ๐ด๐ถ๐ฆ by using the right angle hypotenuse side congruency criterion.
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You might not need to show that level of working in every question.
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But itโs good to demonstrate that it means that this length of ๐ต๐ฆ is the same as the length of ๐ฆ๐ถ.
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Theyโll both be six centimeters.
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This working also proves an important property of isosceles triangles that the median to the base of an isosceles triangle is perpendicular to the base.
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In other words, this line from ๐ด to ๐ต๐ถ connecting at the midpoint ๐ฆ will be perpendicular to the base ๐ต๐ถ.
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Letโs continue with this question and apply the Pythagorean theorem.
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Using the triangle ๐ด๐ต๐ฆ, we can see that thereโs a hypotenuse of 10 and the other two sides will be six and โ.
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So we write 10 squared is equal to six squared plus โ squared.
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Evaluating the squares, 100 is equal to 36 plus โ squared.
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Subtracting 36 from both sides gives us 64 is equal to โ squared.
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We should recognize that 64 is a perfect square.
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So when we take the square root, weโll have โ is equal to eight.
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And the units will be the length units of centimeters.
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In this question, remember that weโre finding the area, not just the perpendicular height.
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So weโll use the area formula now.
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When weโre filling in our values for the base and the height, remember that weโre using this base of 12 centimeters, not six centimeters, and weโre multiplying that by a half and then by eight, the perpendicular height.
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We can simplify before we multiply to give us a value of 48.
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And as weโre working with an area, weโll need square units.
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We can give our answer then that the area of triangle ๐ด๐ต๐ถ is 48 square centimeters.
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We can now summarize what weโve learnt in this video.
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Firstly, we saw that isosceles triangles have two sides equal in length.
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Secondly, we saw some terminology.
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The unequal side in an isosceles triangle is called the base.
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Next, we saw that because isosceles triangles have two sides equal in length, this means that they have two equal angles.
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The base angles will be equal in size.
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And finally, as we saw in our last question, the median to the base of an isosceles triangle is perpendicular to the base.