the bases of isosceles trapeziod abcd are 17 and 25cm and its base angles are 45 degrees. find the height of the trapezoid
To find the height of the isosceles trapezoid, we can follow these steps:
Step 1: Identify the given information:
- Bases of the trapezoid: 17 cm and 25 cm
- Base angles: 45 degrees
- The trapezoid is isosceles, meaning the non-parallel sides are congruent.
Step 2: Draw the trapezoid:
Draw a trapezoid and label the bases AB and CD. Label the non-parallel sides as AD and BC. Also, label the height as h.
Step 3: Identify any congruent sides:
Since the trapezoid is isosceles, AD and BC are congruent.
Step 4: Divide the trapezoid into two right-angled triangles:
Draw a perpendicular line from A to CD, which intersects at a right angle. This perpendicular line is the height of the trapezoid, h. Now, we have two right-angled triangles, △ABC and △ABD.
Step 5: Use the properties of right-angled triangles:
In △ABC, the angle at C is a right angle because AD is perpendicular to CD. Since the base angles of the trapezoid are 45 degrees, the angle at B in △ABC is also 45 degrees.
Step 6: Use trigonometric ratios:
In △ABC, we have the base side BC (17 cm), the angle at B (45 degrees), and we need to find the height h. We can use the trigonometric ratio for tangent (tan):
tan B = BC / h
tan(45 degrees) = 17 / h
1 = 17 / h (Since tan(45 degrees) = 1)
h = 17 cm
Therefore, the height of the isosceles trapezoid is 17 cm.
To find the height of the isosceles trapezoid, we can use the formula for the area of a trapezoid. The formula is A = (a + b) * h / 2, where A is the area, a and b are the lengths of the bases, and h is the height.
In this case, the bases of the trapezoid are given as 17 cm and 25 cm. The height is what we need to find. However, we don't have enough information to directly calculate the height using just the given numbers.
However, it is mentioned that the base angles of the trapezoid are 45 degrees. In an isosceles trapezoid, the base angles are congruent (i.e., they have the same measure). Since we know one of the base angles is 45 degrees, the other base angle must also be 45 degrees.
Now, let's draw the isosceles trapezoid ABCD with the given dimensions.
```
A ------- B
/ \
/ \
D ------------- C
```
Since the base angles are 45 degrees, we can draw perpendiculars from points A and B to the line segment CD, forming two right triangles. Let's label the height of the trapezoid as 'h' and the length of each perpendicular as 'x' (they are the same measure because the trapezoid is isosceles).
```
A ------- B
/ | | \
/ | | \
D ---|---|---|--- C
```
Now, we have formed two right triangles, ADC and BCD. The sum of the base angles of a triangle is always 180 degrees, so in triangle ADC, we have the equation:
45 degrees + 90 degrees + 45 degrees = 180 degrees
Simplifying, 180 degrees = 180 degrees.
We can apply the same logic to triangle BCD.
Since both triangles ADC and BCD are right triangles, we can use trigonometric ratios to find the values of x and h.
For any right triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side.
Using the tangent function, we can write:
tan(45 degrees) = x / h
Since tangent is equal to 1 for a 45-degree angle, the equation becomes:
1 = x / h
x = h
Now, let's substitute the given base lengths into the area formula and solve for the height.
A = (a + b) * h / 2
A = (17 + 25) * h / 2
A = 42 * h / 2
A = 21h
To proceed further and find the value of the height, we need to know the area of the trapezoid or have another piece of information.