Question

the pythagorean theorem can be used to: a. solve quadratic equations b. find the surface area of a sphere c. determine the distance between two points on a coordinate plane d. calculate the circumference of a circle which of the following is not a right triangle? a. a triangle with angles 60°, 60°, and 60° b. a triangle with angles 20°, 70°, and 90° c. a triangle with angles 45°, 45°, and 90° d. a triangle with angles 30°, 60°, and 90°

Explanation:

First Question

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem is $a^{2}+b^{2}=c^{2}$ for a right - triangle with legs $a$ and $b$ and hypotenuse $c$. In a coordinate plane, if we have two points $(x_1,y_1)$ and $(x_2,y_2)$, the distance $d$ between them can be found using the formula $d = \sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}$, which is based on the Pythagorean Theorem. It is not used for solving quadratic equations (quadratic formula or factoring is used for that), finding surface area of a sphere ($4\pi r^{2}$) or calculating circumference of a circle ($2\pi r$).

Second Question

Step1: Recall right - triangle definition

A right - triangle has one angle equal to $90^{\circ}$.

Step2: Analyze each option

• Option a: A triangle with angles $60^{\circ},60^{\circ},60^{\circ}$ is an equilateral triangle and has no $90^{\circ}$ angle.
• Option b: Has a $90^{\circ}$ angle, so it is a right - triangle.
• Option c: Has a $90^{\circ}$ angle, so it is a right - triangle.
• Option d: Has a $90^{\circ}$ angle, so it is a right - triangle.

Answer:

First Question: c. Determine the distance between two points on a coordinate plane
Second Question: a. A triangle with angles $60^{\circ},60^{\circ}$, and $60^{\circ}$