Question 1

A ladder is leaning against a wall, forming a right triangle with the ground. If the ladder is 10 meters long and the base of the ladder is 6 meters from the wall, what is the height at which the ladder touches the wall? Use the Pythagorean Theorem.

Accepted Answer

$$8	ext{ meters}$$

Question 2

Which of the following equations correctly represents the quadratic formula for solving $$ax^2 + bx + c = 0$?

Accepted Answer

$$x = \frac{-b \pm \sqrt{b^2$ - 4ac}}{2a}$$

Question 3

Given two points $$A(2, 3)$$ and $$B(7, 11)$$, what is the distance between them using the distance formula?

Accepted Answer

$$\sqrt{74}$$

Question 4

A triangle has a base of $$12	ext{ cm}$$ and a height of $$5	ext{ cm}. $$What is its area?

Accepted Answer

$$30	ext{ cm}^2$$

Question 5

A construction worker needs to calculate the length of a diagonal brace for a rectangular frame with sides $$9	ext{ ft}$$ and $$12	ext{ ft}. $$What is the length of the brace?

Accepted Answer

$$15	ext{ ft}$$

Question 6

If the quadratic equation $$2x^2 - 4x - 6 = 0$ is solved using the quadratic formula, what are the solutions for x$?

Accepted Answer

$$x = 3,\ x = -1$$

Question 7

A triangle has sides of length $$7	ext{ cm}$$, $$24	ext{ cm}$$, and $$25	ext{ cm}. Is $$this triangle a right triangle?

Accepted Answer

Yes, because $$25^2$$ = $$7^2$$ + $$24^2$$

Question 8

A surveyor needs to find the area of a triangular plot with a base of $$20	ext{ m}$$ and a height of $$8	ext{ m}. $$What is the area?

Accepted Answer

$$80	ext{ m}^2$$

Question 9

Given the points $$(1, 2)$$ and $$(4, 6)$$, what is the distance between them?

Accepted Answer

$$5$$

Question 10

A right triangle has legs of $$9	ext{ m}$$ and $$12	ext{ m}. $$What is the length of the hypotenuse?

Accepted Answer

$$15	ext{ m}$$

Question 11

A triangle has an area of $$42	ext{ cm}^2$$ and a base of $$7	ext{ cm}. $$What is its height?

Accepted Answer

$$12	ext{ cm}$$

Question 12

Which of the following is the correct formula for the area of a triangle?

Accepted Answer

$$A = \frac{h_b b}{2}$$

Question 13

A triangle has a base of $$15	ext{ m}$$ and a height of $$10	ext{ m}. $$What is its area?

Accepted Answer

$$75	ext{ m}^2$$

Question 14

If the distance between $$(0, 0)$$ and $$(6, 8) is $$calculated, what is the result?

Accepted Answer

$$10$$

Question 15

A triangle has sides $$a = 5$$, $$b = 12$$, and $$c = 13. $$Does this triangle satisfy the Pythagorean Theorem?

Accepted Answer

Yes, because $$13^2$$ = $$5^2$$ + $$12^2$$

Question 16

A triangle has an area of $$24	ext{ m}^2$$ and a height of $$8	ext{ m}. $$What is the length of its base?

Accepted Answer

$$6	ext{ m}$$

Question 17

Which formula would you use to find the length between two cities given their coordinates on a map?

Accepted Answer

$$d = \sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$}$$

Question 18

A triangle has a base of $$18	ext{ cm}$$ and a height of $$7	ext{ cm}. $$What is its area?

Accepted Answer

$$63	ext{ cm}^2$$

Question 19

If the quadratic equation $$x^2 - 6x + 9 = 0$ is solved, what are the roots?

Accepted Answer

$$x = 3$$

Question 20

A triangle has a base of $$10	ext{ m}$$ and an area of $$35	ext{ m}^2. $$What is its height?

Accepted Answer

$$7	ext{ m}$$

Essential Algebra and Geometry Formulas for Trigonometry Foundations

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