Question Video: Finding the Side Length in a Triangle Using the Right-Angled Triangle Altitude Theorem | Nagwa Question Video: Finding the Side Length in a Triangle Using the Right-Angled Triangle Altitude Theorem | Nagwa

Question Video: Finding the Side Length in a Triangle Using the Right-Angled Triangle Altitude Theorem Mathematics • Second Year of Preparatory School

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Find the length of the side 𝐴𝐡.

01:28

Video Transcript

Find the length of the side 𝐴𝐡.

To find the length of side 𝐴𝐡 in triangle 𝐴𝐡𝐢, we note first that this is a right triangle at 𝐴 and that 𝐷 is the perpendicular projection of 𝐴 onto side 𝐡𝐢. This being the case, we recall the right triangle altitude theorem, which we can use to find the length of side 𝐴𝐡. This gives us a formula for each of the squares of sides 𝐡𝐴 and 𝐢𝐴. And we see it’s the first of these that we can use to find side length 𝐴𝐡, since this is the same as 𝐡𝐴.

We have 𝐡𝐴 squared equals 𝐡𝐷 multiplied by 𝐡𝐢, where we see that 𝐡𝐢 is equal to the sum of 𝐡𝐷 and 𝐢𝐷. We’re given these lengths in the diagram. 𝐡𝐷 equals 9.6 and 𝐢𝐷 equals 5.4 centimeters. So, we have 𝐡𝐴 squared equal to 9.6 multiplied by the sum of 9.6 and 5.4. That’s 9.6 times 15, which is equal to 144. So, we have 𝐡𝐴 squared equal to 144. And taking the positive square root on both sides, positive since lengths are positive, we have 𝐡𝐴 equal to 12.

Since 𝐡𝐴 is the same as 𝐴𝐡, using the right triangle altitude theorem, we’ve been able to find the length of side 𝐴𝐡, which is equal to 12 centimeters.

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