How to Solve for the Height of a Triangle Using Similar Triangles
Similar triangles, in geometry, are triangles identical in shape to each other, in which their corresponding angles are equal in magnitude to each other and their corresponding sides are proportionate to each other. For instance, a 30-60-90 degree triangle with sides of 1, 2, and sqrt(3) is similar to a 30-60-90 degree triangle with sides of 4, 8, and 4*sqrt(3). You can use the proportionality of the sides of similar triangles to determine the height of a triangle, assuming you know the proportion between the triangles' sides and the height of its similar triangle.
Other People Are Reading
Instructions
-
-
1
Determine the value of the proportion between the sides of the two triangles. For instance, let's say that, in our 30-60-90 degree triangles, the side of length 1, in one triangle, opposite from the 30 degree angle, corresponds to the side of length 4 in the other triangle. Then, the ratio of the two sides equals 1:4.
-
2
Multiply the height of the similar triangle, that you know, by the necessary proportion, to find the height of the other triangle. Let's say that we know the height of the larger triangle, and it equals 3*sqrt(3). Since the sides of the larger triangle are three times longer than the sides of the smaller triangle, multiply 3*sqrt(3) by 1/3, to obtain sqrt(3) for the length of the height of the smaller triangle.
-
-
3
Verify that the height that you calculated is correct. In this case, since we are working with a right triangle, we can verify that our height is correct by plugging it into the Pythagorean Theorem. The Pythagorean Theorem states that a2+b2=c2, in which a and b are the legs of the triangle and c is the hypotenuse. If a=1 and b=sqrt(3) and c=2, then 1^2 +(sqrt3)^2=2^2. 4=4, so our calculation is correct.
-
1
