Question

solve for ( x ). leave your answer in simplest radical form.

Explanation:

Step1: Find the hypotenuse of the lower right triangle

The lower triangle is a right triangle with legs 6 and 7. Using the Pythagorean theorem \( c = \sqrt{a^2 + b^2} \), where \( a = 6 \) and \( b = 7 \).
\[ c = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85} \]
Wait, no, the upper triangle and the lower triangle: Wait, the upper triangle has one leg 6 and the hypotenuse of the lower triangle as the other leg? Wait, no, looking at the diagram, the two right triangles: the lower one has legs 6 and 7, and the upper one has legs 6 and the hypotenuse of the lower one? Wait, no, actually, the two triangles: the lower right triangle (with right angle at the bottom left) has legs 6 and 7, so its hypotenuse is \( \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85} \)? Wait, no, wait the upper triangle has a right angle, and one leg is 6, and the other leg is the hypotenuse of the lower triangle? Wait, no, maybe I misread. Wait, the diagram: two right triangles, sharing a common hypotenuse? Wait, no, the lower triangle: vertical leg 6, horizontal leg 7, right angle at bottom left. The upper triangle: vertical leg 6, horizontal leg (the hypotenuse of the lower triangle), right angle at the left? Wait, no, the upper triangle has a right angle, and one leg is 6, and the other leg is the hypotenuse of the lower triangle? Wait, no, let's re-express.

Wait, the lower triangle: right-angled at the bottom left, legs 6 (vertical) and 7 (horizontal). So its hypotenuse (let's call it \( y \)) is \( y = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85} \)? No, wait, no, the upper triangle is right-angled, with one leg 6 and the other leg \( y \) (the hypotenuse of the lower triangle), and the hypotenuse of the upper triangle is \( x \). Wait, no, actually, the two triangles: the lower one has legs 6 and 7, so hypotenuse \( y = \sqrt{6^2 + 7^2} = \sqrt{85} \). Then the upper triangle has legs 6 and \( y \), so \( x = \sqrt{6^2 + y^2} = \sqrt{36 + 85} = \sqrt{121} = 11 \)? Wait, that can't be. Wait, no, maybe the lower triangle's legs are 6 and 7, and the upper triangle's legs are 6 and the same as the lower triangle's hypotenuse? Wait, no, maybe I made a mistake. Wait, let's check again.

Wait, the diagram: two right triangles. The lower triangle: right angle at the bottom, vertical side 6, horizontal side 7. The upper triangle: right angle at the left, vertical side 6, and the hypotenuse of the lower triangle is the horizontal side of the upper triangle? Wait, no, the two triangles share a common hypotenuse? Wait, no, the upper triangle has a right angle, and one leg is 6, and the other leg is the hypotenuse of the lower triangle. Wait, no, let's use the Pythagorean theorem correctly.

Wait, the lower triangle: legs 6 and 7, so hypotenuse \( d = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85} \). Then the upper triangle: legs 6 and \( d \), so \( x = \sqrt{6^2 + d^2} = \sqrt{36 + 85} = \sqrt{121} = 11 \). Oh! Wait, 36 + 85 is 121, which is 11 squared. So that's correct.

Wait, let's redo:

Step1: Calculate the hypotenuse of the lower right triangle

The lower triangle is a right triangle with legs \( a = 6 \) and \( b = 7 \). By the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85} \]
Wait, no, that's not right. Wait, 6 squared is 36, 7 squared is 49, 36 + 49 is 85, so hypotenuse is \( \sqrt{85} \). Then the upper triangle is a right triangle with legs 6 and \( \sqrt{85} \), so \( x = \sqrt{6^2 + (\sqrt{85})^2} = \sqrt{36 + 85} = \sqrt{121} = 11 \). Ah, right! Because \( (\sqrt{85})^2 = 85 \), so 36 + 85 is 121, square root of 121 is 11. So that's the correct approach.

Step2: Calculate \( x \) using the hypotenuse of the lower triangle

The upper triangle is a right triangle with legs 6 and the hypotenuse of the lower triangle (which we found as \( \sqrt{85} \), but when we square it, it's 85). So:
\[ x = \sqrt{6^2 + (\sqrt{85})^2} = \sqrt{36 + 85} = \sqrt{121} = 11 \]

Answer:

\( x = 11 \)