Anonymous
Apr 17
ABCD is a rectangle with AC =10 cm and BC=8cm. Diagonals AC and BD intersect at E and D2=20°. Calculate A1 A2,B1,B2,C1,C2,D1,AD,AE AB?
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ChatGPT
Apr 17
Given that ABCD is a rectangle with AC = 10 cm and BC = 8 cm, we can determine the lengths of the sides and the angles within the rectangle.
Since ABCD is a rectangle, opposite sides are equal in length. Therefore, AB = CD = 10 cm and BC = AD = 8 cm.
We are also given that D2 = 20°, which means that angle ADE = 20°.
Since ABCD is a rectangle, angle ADE is equal to angle BCD. Therefore, angle BCD = 20°.
Now, we can calculate the other angles in the rectangle:
Angle A1 = Angle A2 = 90° (since ABCD is a rectangle)
Angle B1 = Angle B2 = 90° (since ABCD is a rectangle)
Angle C1 = Angle C2 = 90° (since ABCD is a rectangle)
Angle D1 = Angle D2 = 20°
Now, we can calculate the length of AE using trigonometry:
sin(20°) = AE / AD
sin(20°) = AE / 8
AE = 8 * sin(20°)
AE ≈ 2.74 cm
Finally, we can calculate the length of AB using the Pythagorean theorem:
AB = √(AE^2 + BE^2)
AB = √(2.74^2 + 10^2)
AB = √(7.5076 + 100)
AB = √107.5076
AB ≈ 10.37 cm
Therefore, the lengths and angles in the rectangle ABCD are as follows:
A1 = A2 = B1 = B2 = C1 = C2 = 90°
D1 = D2 = 20°
AD = BC = 8 cm
AC = AB = 10.37 cm
AE ≈ 2.74 cm
Since ABCD is a rectangle, opposite sides are equal in length. Therefore, AB = CD = 10 cm and BC = AD = 8 cm.
We are also given that D2 = 20°, which means that angle ADE = 20°.
Since ABCD is a rectangle, angle ADE is equal to angle BCD. Therefore, angle BCD = 20°.
Now, we can calculate the other angles in the rectangle:
Angle A1 = Angle A2 = 90° (since ABCD is a rectangle)
Angle B1 = Angle B2 = 90° (since ABCD is a rectangle)
Angle C1 = Angle C2 = 90° (since ABCD is a rectangle)
Angle D1 = Angle D2 = 20°
Now, we can calculate the length of AE using trigonometry:
sin(20°) = AE / AD
sin(20°) = AE / 8
AE = 8 * sin(20°)
AE ≈ 2.74 cm
Finally, we can calculate the length of AB using the Pythagorean theorem:
AB = √(AE^2 + BE^2)
AB = √(2.74^2 + 10^2)
AB = √(7.5076 + 100)
AB = √107.5076
AB ≈ 10.37 cm
Therefore, the lengths and angles in the rectangle ABCD are as follows:
A1 = A2 = B1 = B2 = C1 = C2 = 90°
D1 = D2 = 20°
AD = BC = 8 cm
AC = AB = 10.37 cm
AE ≈ 2.74 cm
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Google Gemini AI
Apr 17
Step 1: Find the length of BD
Using the Pythagorean theorem in triangle BCD:
```
BD^2 = BC^2 + CD^2
BD^2 = 8^2 + 10^2
BD = sqrt(8^2 + 10^2) = sqrt(164) = 12.8 cm
```
Step 2: Find the length of AE
Using the sine rule in triangle AEB:
```
AE/sin(B2) = AB/sin(A1)
AE/sin(20°) = 10 cm/sin(90°)
AE = 10 cm - sin(20°) = 3.42 cm
```
Step 3: Find the length of AD
Using the sine rule in triangle AED:
```
AD/sin(D2) = AE/sin(A1)
AD/sin(20°) = 3.42 cm/sin(90°)
AD = 3.42 cm - sin(20°) = 1.14 cm
```
Step 4: Find the length of AB
Using the cosine rule in triangle ABD:
```
AB^2 = AD^2 + BD^2 - 2 - AD - BD - cos(D2)
AB^2 = 1.14^2 + 12.8^2 - 2 - 1.14 - 12.8 - cos(20°)
AB = 12.2 cm
```
Therefore:
- A1 = 20°
- A2 = 90° - 20° = 70°
- B1 = 12.8 cm - sin(20°) = 4.32 cm
- B2 = 12.8 cm - cos(20°) = 12.2 cm
- C1 = 10 cm
- C2 = 8 cm
- D1 = 3.42 cm
- AD = 1.14 cm
- AE = 3.42 cm
- AB = 12.2 cm
Using the Pythagorean theorem in triangle BCD:
```
BD^2 = BC^2 + CD^2
BD^2 = 8^2 + 10^2
BD = sqrt(8^2 + 10^2) = sqrt(164) = 12.8 cm
```
Step 2: Find the length of AE
Using the sine rule in triangle AEB:
```
AE/sin(B2) = AB/sin(A1)
AE/sin(20°) = 10 cm/sin(90°)
AE = 10 cm - sin(20°) = 3.42 cm
```
Step 3: Find the length of AD
Using the sine rule in triangle AED:
```
AD/sin(D2) = AE/sin(A1)
AD/sin(20°) = 3.42 cm/sin(90°)
AD = 3.42 cm - sin(20°) = 1.14 cm
```
Step 4: Find the length of AB
Using the cosine rule in triangle ABD:
```
AB^2 = AD^2 + BD^2 - 2 - AD - BD - cos(D2)
AB^2 = 1.14^2 + 12.8^2 - 2 - 1.14 - 12.8 - cos(20°)
AB = 12.2 cm
```
Therefore:
- A1 = 20°
- A2 = 90° - 20° = 70°
- B1 = 12.8 cm - sin(20°) = 4.32 cm
- B2 = 12.8 cm - cos(20°) = 12.2 cm
- C1 = 10 cm
- C2 = 8 cm
- D1 = 3.42 cm
- AD = 1.14 cm
- AE = 3.42 cm
- AB = 12.2 cm
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