Question

a ladder leans against a wall, forming a right triangle. the ladder is 13 ft long and the base is 5 ft from the wall. what is the height of the ladder on the wall?
a. 10 ft
b. 15 ft
c. 12 ft
d. 7 ft

in a triangle, if one side is divided into segments of 6 m and 9 m, and a line parallel to the base divides the other side into segments 8 m and x, what is the value of x?
a. 16 m
b. 10 m
c. 12 m
d. 14 m

Explanation:

Step1: Apply Pythagorean theorem for first - question

In a right - triangle formed by the ladder, wall and ground, let the length of the ladder be the hypotenuse $c = 13$ ft and the base distance from the wall be $a = 5$ ft. We want to find the height $b$ on the wall. By the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, so $b=\sqrt{c^{2}-a^{2}}$.

Step2: Calculate the height

Substitute $c = 13$ and $a = 5$ into the formula: $b=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12$ ft.

Step3: Apply the basic proportionality theorem for second - question

The basic proportionality theorem (Thales' theorem) states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. So, $\frac{6}{9}=\frac{8}{x}$.

Step4: Solve for $x$

Cross - multiply: $6x=9\times8$. Then $6x = 72$, and $x=\frac{72}{6}=12$ m.

Answer:

1. C. 12 ft
2. C. 12 m