The base of triangle ABC is one half the altitude.The altitude of triangle DEF is one half its base. If both triangles are equal in area, what is the ratio of the altitude of triangle ABC to the altitude of triangle DEF ?
a. 1:4
b. 1:2
c. 1:1
d. 2:1
e. 4:1
please answer and explain
Let's assume the base of triangle ABC is represented by "b" and the altitude is represented by "h".
Given that the base of triangle ABC is one half the altitude, we can write:
b = 1/2 * h
Now, let's consider triangle DEF. The altitude is represented by "x" and the base is represented by "y".
Given that the altitude of triangle DEF is one half its base, we can write:
x = 1/2 * y
We are also given that both triangles are equal in area.
The area of a triangle can be calculated using the formula A = 1/2 * base * height.
For triangle ABC, the area is given as A_ABC = 1/2 * b * h
For triangle DEF, the area is given as A_DEF = 1/2 * y * x
Since the areas of both triangles are equal, we can equate these formulas:
1/2 * b * h = 1/2 * y * x
Substituting the values of b and x from the given information, we have:
1/2 * (1/2 * h) * h = 1/2 * y * (1/2 * y)
Simplifying this equation, we get:
1/4 * h^2 = 1/4 * y^2
Canceling out the common factors, we have:
h^2 = y^2
Taking the square root of both sides, we have:
h = y
Therefore, the ratio of the altitude of triangle ABC to the altitude of triangle DEF is 1:1.
Hence, the correct answer is option c. 1:1.
To find the ratio of the altitude of triangle ABC to the altitude of triangle DEF, we need to compare their heights.
Let's assume the base of triangle ABC is denoted by b and the altitude of triangle DEF is denoted by h.
According to the given information:
1) The base of triangle ABC is one half the altitude. So, the base of triangle ABC is b = (1/2)h.
2) The altitude of triangle DEF is one half its base. So, the altitude of triangle DEF is h = (1/2)b.
Now, let's find the areas of both triangles. The area of a triangle is given by the formula A = (1/2) * base * height.
For triangle ABC:
Area of triangle ABC = (1/2) * b * h
= (1/2) * (1/2)h * h
= (1/4)h^2
For triangle DEF:
Area of triangle DEF = (1/2) * b * h
= (1/2) * (1/2)b * b
= (1/4)b^2
Given that both triangles have equal areas, we can equate their areas:
(1/4)h^2 = (1/4)b^2
Dividing both sides by (1/4):
h^2 = b^2
Taking the square root of both sides:
h = b
Since b = (1/2)h, we can substitute this value into the equation:
h = (1/2)h
Simplifying:
2h = h
Finally, we can conclude that the altitude of triangle ABC (h) is equal to the altitude of triangle DEF (h).
Therefore, the ratio of the altitude of triangle ABC to the altitude of triangle DEF is 1:1, which means the correct answer is c. 1:1.