8:06 Score:0/6 Wors Problems, Similar Triangles (Level 2) Question A guy wire runs from the top of a cell tower to a metal stake in the ground. Grace places a 6-foot tall pole to support the guy wire. After placing the pole, Grace measures the distance from the stake to the pole to be 7 ft . She then measures the distance from the pole to the tower to be 19 ft. Find the length of the guy wire, to the nearest foot. Answer Attempt 1 out of 2

To determine the length of the guy wire, we can use the concept of similar triangles. Here's a step-by-step breakdown: 1. **Visualizing the Problem:** - Let's denote: - **T** as the top of the cell tower. - **S** as the stake in the ground where the guy wire is anchored. - **P** as the point where Grace places the 6-foot pole. - The distances given are: - From the stake (**S**) to the pole (**P**) horizontally: **7 feet**. - From the pole (**P**) to the base of the tower horizontally: **19 feet**. - Therefore, the total horizontal distance from the stake to the tower is **7 + 19 = 26 feet**. 2. **Setting Up the Similar Triangles:** - The height of the pole (**P**) is **6 feet**, and let’s denote the height of the tower (**T**) as **h** feet. - At point **P**, the wire's height is 6 feet. At the base (**S**), the height is 0 feet, and at the tower (**T**), it's **h** feet. - Since the triangles formed are similar, the ratios of corresponding sides are equal: \[ \frac{h - 6}{h} = \frac{7}{26} \] 3. **Solving for the Height of the Tower (\(h\)):** \[ h - 6 = \frac{7h}{26} \] \[ 26h - 156 = 7h \] \[ 19h = 156 \] \[ h = \frac{156}{19} \approx 8.21 \text{ feet} \] 4. **Calculating the Length of the Guy Wire:** - The guy wire forms the hypotenuse of a right triangle with the horizontal distance of **26 feet** and the tower's height of approximately **8.21 feet**. - Using the Pythagorean theorem: \[ \text{Length of guy wire} = \sqrt{8.21^2 + 26^2} \approx \sqrt{67.6 + 676} \approx \sqrt{743.6} \approx 27.26 \text{ feet} \] - Rounding to the nearest whole number gives **27 feet**. **Answer:** 27