QuestionProve that given . Find if is right-angled and the area of .
Studdy Solution
STEP 1
Assumptions1. and are straight lines.
. .
3. and .
STEP 2
To prove that , we need to show that their corresponding angles are equal. We are already given that .
STEP 3
Since and are straight lines, we know that and .
STEP 4
This implies that and .
STEP 5
But we know that , so we can say that .
STEP 6
We also know that because they are vertically opposite angles.
STEP 7
So, we have , and . Hence, by the Angle-Angle-Angle (AAA) similarity criterion.
STEP 8
Now, let's check if is a right-angled triangle. For a triangle to be right-angled, the square of the longest side must be equal to the sum of the squares of the other two sides (Pythagorean theorem).
STEP 9
We can write this as .
STEP 10
Substitute the given values into the equation.
STEP 11
Calculate the squares of the sides.
STEP 12
Check if the equation holds true.
STEP 13
Since the equation holds true, is a right-angled triangle.
STEP 14
Now, let's find the area of quadrilateral . We can do this by finding the areas of and and then subtracting the area of from the area of .
STEP 15
The area of a triangle is given by the formula .
STEP 16
The area of is .
STEP 17
Substitute the given values into the equation.
STEP 18
Calculate the area of .
STEP 19
Since , their areas are proportional to the squares of their corresponding sides. Therefore, the ratio of the area of to the area of is the square of the ratio of their corresponding sides.
STEP 20
We can write this as \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEC} = \left(\frac{AB}{DE}\right)^.
STEP 21
Substitute the given values into the equation.
\frac{1500}{\text{Area of } \triangle DEC} = \left(\frac{40}{16}\right)^
STEP 22
olve the equation for the area of .
\text{Area of } \triangle DEC = \frac{1500}{\left(\frac{40}{16}\right)^}
STEP 23
Calculate the area of .
\text{Area of } \triangle DEC =600 \, cm^
STEP 24
Now, subtract the area of from the area of to get the area of quadrilateral .
STEP 25
Substitute the calculated areas into the equation.
STEP 26
Calculate the area of .
\text{Area of } ABDE =900 \, cm^The area of quadrilateral is 900 \, cm^.
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