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In the given figure, π΄π΅ equals 35, π΄πΆ equals 30, and πΆπ· equals 12.
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If π΅π· equals π₯ plus 10, what is the value of π₯?
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Weβve been given a diagram of a triangle and the lengths of various lines within this triangle.
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Letβs first add this information to the diagram.
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The question asked us to find the value of π₯, which forms part of the expression for π΅π·.
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Letβs think about how to approach this problem.
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The line π΄π· is a bisector of the angle πΆπ΄π΅.
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We can see this because the two parts of the angle have each been marked with a single blue arc, indicating that they are equal.
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Therefore, we need to approach this problem using facts about angle bisectors.
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The angle bisector divides the opposite side of the triangle πΆπ΅ into two parts, πΆπ· and π·π΅.
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The ratio between the lengths of these two parts is the same as the ratio of the lengths of the other two sides of the triangle.
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Or, in other words, for this triangle, the ratio we get when we divide by π΅π· by πΆπ· is the same as the ratio we get when we divide π΄π΅ by π΄πΆ.
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In each case, this is the pink side divided by the green side.
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We can substitute in the values or, in the case of π΅π·, the expression for each of these sides to give an equation that we can solve to find the value of π₯.
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π΅π· over πΆπ· becomes π₯ plus 10 over 12.
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π΄π΅ over π΄πΆ becomes 35 over 30.
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This fraction can be simplified by dividing both the numerator and denominator by five to give a simplified fraction of seven over six.
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Now letβs think about how to solve this equation.
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We have a 12 in the denominator of one fraction and a six in the denominator of the other.
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Multiplying both sides of the equation by 12 will eliminate both these denominators.
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The 12 that now appears in the numerator on the right-hand side will cancel with the six in the denominator to give an overall factor of two.
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So weβre left with π₯ plus 10 is equal to seven multiplied by two, which is 14.
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The final step in solving this equation is we need to subtract 10 from both sides.
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This gives π₯ is equal to four.
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So we found the value of π₯.
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Remember, the key fact that we used in this question is that an angle bisector divides the opposite side of a triangle in the same ratio as the ratio that exists between the other two sides of the triangle.