 Suggested languages for you:

Europe

Problem 696

# Show that the points $$\mathrm{A}(2,-2), \mathrm{B}(-8,4)$$, and $$\mathrm{C}(5,3)$$ are the vertices of a right triangle and find its area.

Expert verified
The given points $\mathrm{A}(2,-2), \mathrm{B}(-8,4)$, and $\mathrm{C}(5,3)$ form a right triangle with the right angle at point $\mathrm{B}$. The area of the triangle is $$34$$ square units.
See the step by step solution

## Step 1: Find side lengths AB, BC, and AC using the distance formula.

The distance between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ in a plane can be found using the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$. Let's find the side lengths for the triangle formed by points $$\mathrm{A}(2,-2), \mathrm{B}(-8,4)$$, and $$\mathrm{C}(5,3)$$. Length of AB: $AB = \sqrt{(-8 - 2)^2 + (4 - (-2))^2} = \sqrt{(-10)^2 + (6)^2} = \sqrt{136}$ Length of BC: $BC = \sqrt{(5 - (-8))^2 + (3 - 4)^2} = \sqrt{(13)^2 + (-1)^2} = \sqrt{170}$ Length of AC: $AC = \sqrt{(5 - 2)^2 + (3 - (-2))^2} = \sqrt{(3)^2 + (5)^2} = \sqrt{34}$

## Step 2: Check if the Pythagorean theorem holds true for the side lengths.

We will check if any permutation of the side lengths satisfies the Pythagorean theorem, which states that for a right triangle with sides of length a, b, and c, where c is the longest side, then $$a^2 + b^2 = c^2$$. Check the sum of the squares of the side lengths: $AB^2 + BC^2 = 136 + 170 = 306\\ AB^2 + AC^2 = 136 + 34 = 170\\ BC^2 + AC^2 = 170 + 34 = 204$ We can see that $$AB^2 + AC^2 = 170$$, which is equal to the square of side BC. Therefore, the given points form a right triangle, with the right angle at point $$\mathrm{B}$$.

## Step 3: Find the area of the right triangle.

Since we've determined that the triangle is indeed a right triangle, with the right angle at point $$\mathrm{B}$$, we can now find its area using the following formula: Area = $$\frac{1}{2}\times$$ base $$\times$$ height. In our case, the base is the side AB and the height is the side AC: Area = $$\frac{1}{2} \times \sqrt{136} \times \sqrt{34}$$. To simplify, we can factor out the square root of 2 from both sides: Area = $$\frac{1}{2} \times \sqrt{2^2 \cdot 34} \times \sqrt{2 \cdot 17}$$. Area = $$\frac{1}{2} \times 2\sqrt{34} \times \sqrt{34}$$. Thus, the area of the right triangle is $$34$$ square units.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features ## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner 