Question

7.3.ps-15 metalworking a machine in a factory cuts out four triangular sheets of metal. which of the triangles shown on the right are right triangles? which triangles are right triangles? select all that apply. a. triangle 1 b. triangle 2 c. triangle 4 d. triangle 3 triangle side lengths (in.) triangle 1: 18, 21, √765 triangle 2: 19, 20, √463 triangle 3: 15, 24, √461 triangle 4: 16, 22, √740

Explanation:

To determine if a triangle is a right triangle, we use the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(c\) is the longest side (hypotenuse).

Step 1: Analyze Triangle 1

Sides: \(18\), \(21\), \(\sqrt{765}\)
Calculate \(18^2 + 21^2\):
\(18^2 = 324\), \(21^2 = 441\)
\(324 + 441 = 765\)
\((\sqrt{765})^2 = 765\)
So, \(18^2 + 21^2 = (\sqrt{765})^2\). Triangle 1 is a right triangle.

Step 2: Analyze Triangle 2

Sides: \(19\), \(20\), \(\sqrt{463}\)
Calculate \(19^2 + 20^2\):
\(19^2 = 361\), \(20^2 = 400\)
\(361 + 400 = 761\)
\((\sqrt{463})^2 = 463\)
\(761
eq 463\). Triangle 2 is not a right triangle.

Step 3: Analyze Triangle 3

Sides: \(15\), \(24\), \(\sqrt{461}\)
Calculate \(15^2 + 24^2\):
\(15^2 = 225\), \(24^2 = 576\)
\(225 + 576 = 801\)
\((\sqrt{461})^2 = 461\)
\(801
eq 461\). Triangle 3 is not a right triangle.

Step 4: Analyze Triangle 4

Sides: \(16\), \(22\), \(\sqrt{740}\)
Calculate \(16^2 + 22^2\):
\(16^2 = 256\), \(22^2 = 484\)
\(256 + 484 = 740\)
\((\sqrt{740})^2 = 740\)
So, \(16^2 + 22^2 = (\sqrt{740})^2\). Triangle 4 is a right triangle.

Answer:

A. Triangle 1, D. Triangle 4