Question

a ladder reaches a height of 15 feet on a wall. the base is 9 feet from the wall. how long is the ladder? a phone pole casts a shadow that is 18 feet long. the pole is 17 feet tall. what is the distance from the top of the pole to the end of the shadow? what is the distance between the points (3,5) and (10,9)? what is the distance between the points (-4,-3) and (0,5)?

Explanation:

Step1: Solve the ladder problem

Use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 15\), \(b=9\).

$$c=\sqrt{15^{2}+9^{2}}=\sqrt{225 + 81}=\sqrt{306}$$

Step2: Solve the phone - pole shadow problem

Use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 17\), \(b = 18\).

$$c=\sqrt{17^{2}+18^{2}}=\sqrt{289+324}=\sqrt{613}$$

Step3: Solve the distance between \((3,5)\) and \((10,9)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), where \(x_1 = 3,y_1 = 5,x_2=10,y_2 = 9\).

$$d=\sqrt{(10 - 3)^{2}+(9 - 5)^{2}}=\sqrt{49 + 16}=\sqrt{65}$$

Step4: Solve the distance between \((-4,-3)\) and \((0,5)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), where \(x_1=-4,y_1=-3,x_2 = 0,y_2 = 5\).

$$d=\sqrt{(0+4)^{2}+(5 + 3)^{2}}=\sqrt{16+64}=\sqrt{80}$$

Answer:

• For the ladder problem: \(\sqrt{306}\)
• For the phone - pole shadow problem: \(\sqrt{613}\)
• For the distance between \((3,5)\) and \((10,9)\): \(\sqrt{65}\)
• For the distance between \((-4,-3)\) and \((0,5)\): \(\sqrt{80}\)