Question Video: Finding Unknown Lengths in a Triangle Using the Similarity between Triangles | Nagwa Question Video: Finding Unknown Lengths in a Triangle Using the Similarity between Triangles | Nagwa

Question Video: Finding Unknown Lengths in a Triangle Using the Similarity between Triangles Mathematics

Find the value of π₯.

03:10

Video Transcript

Find the value of π₯.

Here, we have a large triangle π΄π΅πΆ, and inside that triangle are two smaller triangles. Weβre trying to find the value of π₯. And the only side length that has π₯ in it is side π·πΆ, so we need to decide what to set that equal to. The other two side lengths they give us is 22 centimetres for side π΄π· and 22 centimetres for side π·π΅; those are congruent and that will actually be helpful. Because looking at triangle π΄π·π΅, now we can see that this triangle is isosceles. And in an isosceles triangle, the base angles are congruent.

Angle π· is going to be 90 degrees for triangle π΄π·π΅ because the angle that is adjacent to it β right next to it β is 90 degrees. And they make a straight line, so they should add up to 180. So angle π· is 90 and this is important. In a triangle, all three angles add up to 180 degrees. So if we know angle π· for this triangle is already 90, weβve already used 90 degrees up. So now the remainder of 90 degrees needs to be split evenly between angle π΄ and angle π΅, forcing each of them to be 45 degrees.

The reason this is helpful is looking at angle π΅. Angle π΅ we can see that some of it is 45 degrees and the other part of it we donβt know. However, we know the entire angle π΅ for the large triangle is 90 degrees. So if the portion that goes with the blue triangle, triangle π΄π·π΅, is 45 degrees, this portion must also be 45 degrees because 45 plus 45 is 90.

So now focusing on triangle πΆπ·π΅, we have a 90-degree angle, a 45-degree angle, and inside, a triangle all angles add to 180 degrees. This forces angle πΆ to be 45 degrees as well. Because 45 degrees plus 45 degrees plus 90 degrees is equal to 180 degrees. So the two 45 degrees are equal. This makes it an isosceles triangle. These are the base angles and the sides across from the base angles are the legs, which are congruent.

So if side π·π΅ is 22 centimetres, then side π·πΆ is 22 centimetres. Therefore, we can set four π₯ plus three equal to 22. The first step to solve for π₯ would be to subtract three from both sides of the equation. On the left, the threes cancel and on the right 22 minus three is 19. Now I need to divide both sides by four. On the left, the fours cancel and on the right, we get 19 fourths. This means the value of π₯ is equal to 19 fourths.

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