Find the area of the triangle 𝐴𝐵𝐶 correct to one decimal place.
From the diagram, we can see that triangle 𝐴𝐵𝐶 is a right-angled triangle. To find the area of a right-angled triangle, we know that we need to multiply the base by the perpendicular height and divide by two. The base of the triangle has been given to us. It’s 16.1 centimetres. But the height hasn’t.
So before we can find the area of the triangle, we need to calculate its height, the side 𝐴𝐵. How are we going to do this? Well we have a right-angled triangle. And we’ve been given the lengths of two of its sides. We want to find the length of the third side. We can use the Pythagorean theorem to do this.
The Pythagorean theorem tells us that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side, the hypotenuse. If the two shorter sides are labelled 𝑎 and 𝑏 and the hypotenuse is labelled 𝑐, the Pythagorean theorem can be written as 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. Let’s substitute the values of 𝑎 and 𝑐 in this triangle.
We now have the equation 16.1 squared plus 𝑏 squared is equal to 28.9 squared. And this is an equation, we can solve in order to find the length of side 𝑏. Evaluating 16.1 squared and 28.9 squared, gives 259.21 plus 𝑏 squared is equal to 835.21 subtracting 259.21 from both sides of this equation gives 𝑏 squared is equal to 576. Next, we need to take the square root of each side.
So we have that 𝑏 is equal to the square root of 576. And in fact, this is exactly equal to 24. So now we know that the third side of the triangle, the perpendicular height, is 24 centimetres. Now we have all the information we need to be able to calculate the area of the triangle.
So the calculation for the area: base times height divided by two becomes 16.1 multiplied by 24 divided by two. This gives 193.2. Now the question asked for the area correct to one decimal place. And in fact this value is already to the correct degree of accuracy. So we have the area of triangle 𝐴𝐵𝐶, now with its associated units, is 193.2 centimetres squared.