Video: Writing and Solving Quadratic Equations Involving Pythagorean’s Theorem

A right triangle has side lengths of (2π‘₯) cm, (π‘₯ + 5) cm, and (2π‘₯ + 4) cm. Find the value of π‘₯ and calculate the perimeter and area of the triangle.

06:08

Video Transcript

A right triangle has side lengths of two π‘₯ centimeters, π‘₯ plus five centimeters, and two π‘₯ plus four centimeters. Find the value of π‘₯ and calculate the perimeter and area of the triangle.

We will begin by sketching the right triangle. We are told that the triangle has side lengths two π‘₯, π‘₯ plus five, and two π‘₯ plus four, where each of these is measured in centimeters. And convention dictates that the third side is the hypotenuse or longest side. As we are dealing with a right triangle, we can use the Pythagorean theorem. This states that π‘Ž squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side or hypotenuse.

Substituting in the values from our triangle, we have two π‘₯ squared plus π‘₯ plus five squared is equal to two π‘₯ plus four squared. Two π‘₯ multiplied by two π‘₯ is equal to four π‘₯ squared. For the other two terms, we will need to distribute the parentheses, otherwise known as expanding the brackets. We begin by squaring π‘₯ plus five, and we can use the FOIL method. Multiplying the first terms gives us π‘₯ squared. Multiplying the outer terms, we have five π‘₯. When multiplying the inner terms, we also have five π‘₯. Finally, multiplying the last terms gives us 25. Collecting like terms, this simplifies to π‘₯ squared plus 10π‘₯ plus 25. π‘₯ plus five all squared is therefore equal to π‘₯ squared plus 10π‘₯ plus 25.

We can repeat this process when squaring two π‘₯ plus four. Two π‘₯ plus four multiplied by two π‘₯ plus four is equal to four π‘₯ squared plus eight π‘₯ plus eight π‘₯ plus 16. This simplifies to four π‘₯ squared plus 16π‘₯ plus 16. We now have the equation four π‘₯ squared plus π‘₯ squared plus 10π‘₯ plus 25 is equal to four π‘₯ squared plus 16π‘₯ plus 16. We can subtract four π‘₯ squared from both sides of the equation. We can then subtract 16π‘₯ and 16 from both sides of the equation, giving us π‘₯ squared minus six π‘₯ plus nine is equal to zero.

We now have a quadratic equation equal to zero, which we can solve by factoring. π‘₯ squared minus six π‘₯ plus nine is equal to π‘₯ minus three multiplied by π‘₯ minus three. This is because the sum of negative three and negative three is negative six and their product is nine. For the factored form to be equal to zero, then one of the factors must equal zero. As both of our parentheses are π‘₯ minus three, we know that π‘₯ minus three must equal zero. Adding three to both sides of this equation gives us a value of π‘₯ equal to three. We have now answered the first part of the question. π‘₯ is equal to three.

We were also asked to calculate the perimeter and area of the triangle. In order to do this, we will substitute π‘₯ equals three back in to the three expressions for the lengths of our triangle. The three lengths in centimeters were two π‘₯, π‘₯ plus five, and two π‘₯ plus four. As π‘₯ is equal to three, we have two multiplied by three. The first length of our triangle is equal to six centimeters. Next, we have three plus five. The second side length of our triangle is eight centimeters. Finally, we have two multiplied by three plus four. Two multiplied by three is equal to six and adding four to this gives us 10. Therefore, the length of the hypotenuse is 10 centimeters.

We have a right triangle with side lengths six centimeters, eight centimeters, and 10 centimeters. We know that to calculate the perimeter of any shape, we simply add the lengths around the outside. Six plus eight plus 10 is equal to 24. As the perimeter is a measure of length, this is equal to 24 centimeters. We know that the area of any triangle is equal to its base multiplied by its height divided by two. In this question, we need to multiply eight by six and then divide our answer by two or halve it. Eight multiplied by six is 48, and dividing this by two gives us 24. As area is measured in square units, the area of our triangle is 24 square centimeters.

If a right triangle has side lengths of two π‘₯ centimeters, π‘₯ plus five centimeters, and two π‘₯ plus four centimeters, then the value of π‘₯ is equal to three, the perimeter of the triangle is 24 centimeters, and the area is 24 square centimeters.

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