Questions: 5. In triangle PQR, angle P is a right angle. Find the perimeter of triangle PQR to the nearest tenth. 6. A right rectangular prism has a length, width, and height that are all equal. The volume of the prism is 343 cubic meters. Find the surface area.

5. In triangle PQR, angle P is a right angle. Find the perimeter of triangle PQR to the nearest tenth.
6. A right rectangular prism has a length, width, and height that are all equal. The volume of the prism is 343 cubic meters. Find the surface area.

Transcript text: 5. In $\triangle P Q R, \angle P$ is a right angle. Find the perimeter of $\triangle P Q R$ to the nearest tenth. 6. A right rectangular prism has a length, width, and height that are all equal. The volume of the prism is 343 cubic meters. Find the surface area.

Solution

Solution Steps

Step 1: Find the length of XY

Since RX is parallel to PY, triangles PRX and PQY are similar. Therefore, the ratio of corresponding sides is equal. PX/PY = RX/QY. Substituting the given values, we get 5/(5+x) = 4/(4+6). Simplifying, we have 5/(5+x) = 4/10. Cross-multiplying yields 50 = 20 + 4x. Solving for x, we find x=7.5. Thus, XY = x = 7.5.

Step 2: Find the length of PR

Use the Pythagorean theorem on triangle PXR. PR² = PX² + RX² Substituting known values, PR² = 5² + 4² = 25 + 16 = 41. Thus, PR = √41.

Step 3: Find the length of PQ

Use the similarity of triangles PXR and PYQ. PQ/PR = PY/PX. Substituting and solving for PQ: PQ/√41 = (5+7.5)/5. PQ = (12.5/5)√41 = 2.5√41.

Step 4: Find the length of QR

Use the Pythagorean theorem on triangle PQR. QR² = PQ² + PR² = (2.5√41)² + (√41)² = 6.25 * 41 + 41 = 7.25 * 41 = 297.25. Therefore, QR = √297.25 ≈ 17.2.

Step 5: Calculate the perimeter of triangle PQR

Perimeter = PR + PQ + QR ≈ √41 + 2.5√41 + 17.2 ≈ 6.4 + 16 + 17.2 ≈ 39.6.