WEBVTT
00:00:01.743 --> 00:00:08.033
In the given figure, ππ equals three π₯ minus seven and ππ equals two π₯ minus two.
00:00:08.263 --> 00:00:09.603
Find ππ.
00:00:10.433 --> 00:00:15.513
We can begin by filling in the given information about the lengths onto the diagram.
00:00:15.973 --> 00:00:24.663
Weβre not told any information about this figure other than the fact that there is a pair of right angles and thereβs also another pair of congruent angles.
00:00:25.153 --> 00:00:28.713
So, letβs check if these two triangles are congruent.
00:00:29.273 --> 00:00:36.103
Letβs start by noting that we have two right angles, angle π½ππ and angle π½ππ.
00:00:36.433 --> 00:00:38.943
So, we have a pair of congruent angles.
00:00:39.473 --> 00:00:46.273
We can see that angle ππ½π and angle ππ½π from the diagram are also congruent.
00:00:46.913 --> 00:00:52.703
Weβre given the lengths for ππ and ππ, but weβre not told that these are congruent.
00:00:53.123 --> 00:00:57.433
And weβre not given any information about the line π½π or π½π.
00:00:57.793 --> 00:01:05.303
We can see, however, that the line π½π is common to both triangles, which means that we have a pair of congruent sides.
00:01:05.753 --> 00:01:17.913
As we found two angles and a nonincluded side, we could say that triangle π½ππ and triangle π½ππ are congruent using the angle-angle-side or AAS rule.
00:01:18.323 --> 00:01:21.703
Letβs see if this will help us to find the length ππ.
00:01:22.063 --> 00:01:32.263
Using the fact that these triangles are congruent, we know that the length ππ in triangle π½ππ is congruent with the length ππ in triangle π½ππ.
00:01:32.583 --> 00:01:37.293
And so, two π₯ minus two must be equal to three π₯ minus seven.
00:01:37.543 --> 00:01:40.103
So, weβll need to solve to find the value of π₯.
00:01:40.433 --> 00:01:49.273
If we rewrite this so that we have our higher coefficient of π₯ on the left-hand side, weβll have three π₯ minus seven equals two π₯ minus two.
00:01:49.703 --> 00:01:57.063
Subtracting two π₯ from both sides of the equation, weβll have π₯ minus seven equals negative two.
00:01:57.413 --> 00:02:04.253
We can then add seven to both sides of the equation, so π₯ equals negative two plus seven.
00:02:04.553 --> 00:02:06.783
And therefore, π₯ is equal to five.
00:02:07.203 --> 00:02:10.733
It can be easy to stop at this point and think that weβve found the answer.
00:02:10.903 --> 00:02:12.743
However, we werenβt asked for π₯.
00:02:12.773 --> 00:02:14.723
We were asked for the length ππ.
00:02:15.283 --> 00:02:18.763
We were given that ππ equals three π₯ minus seven.
00:02:18.963 --> 00:02:27.493
So, we plug in our value of π₯ equals five into this equation, which gives us ππ equals three times five minus seven.
00:02:27.663 --> 00:02:32.553
And as three times five is 15, weβll have 15 minus seven, which is eight.
00:02:32.983 --> 00:02:39.663
And so, we have our answer ππ equals eight, which we found by proving that the two triangles were congruent.