A parallelogram has a 70 degree angle and sides 6 cm and 10 cm long. How long are its diagonals?
I assume you have had trig. Draw the figure. if it has a 70 degree acute angle, the other angle is 110.
Use the law of cosines:
shortdiag^2=6^2+7^2-2*6*7Cos70
longdiag^2=6^2+7^2-2*6*7Cos110
To find the lengths of the diagonals of a parallelogram with given angle and side lengths, we can use the properties of parallelograms.
Step 1: Recall that a parallelogram has opposite sides that are equal in length and opposite angles that are equal in measure.
Step 2: Since the given angle is 70 degrees, we know that the opposite angle in the parallelogram is also 70 degrees.
Step 3: To find the lengths of the diagonals, we need to use the given side lengths and angle measures. Let's denote the diagonals as AC and BD.
Step 4: Since the diagonals of a parallelogram bisect each other, AC and BD will intersect at their midpoints. Let's denote this point of intersection as O.
Step 5: Using the properties of parallelograms, we can form two congruent right triangles, ACO and BDO, by drawing perpendiculars from point O to sides AD and BC.
Step 6: Let's apply the trigonometric ratios to find the lengths of the diagonals.
In triangle ACO:
- Let x be the length of the diagonal AC.
- The adjacent side to the angle of 70 degrees is 6 cm.
- Therefore, we have cos(70) = 6 / x.
In triangle BDO:
- Let y be the length of the diagonal BD.
- The adjacent side to the angle of 70 degrees is 10 cm.
- Therefore, we have cos(70) = 10 / y.
Step 7: Simplify the trigonometric equations:
- For triangle ACO, we have: cos(70) = 6 / x.
Rearranging the equation, we get: x = 6 / cos(70).
- For triangle BDO, we have: cos(70) = 10 / y.
Rearranging the equation, we get: y = 10 / cos(70).
Step 8: Calculate the values of x and y:
- Using a calculator, calculate cos(70): cos(70) ≈ 0.3420.
- Substitute the value of cos(70) into the equations for x and y:
x = 6 / 0.3420 ≈ 17.54 cm.
y = 10 / 0.3420 ≈ 29.24 cm.
Step 9: Therefore, the length of diagonal AC is approximately 17.54 cm, and the length of diagonal BD is approximately 29.24 cm.
Note: The lengths of the diagonals in a parallelogram are not always equal, unless the parallelogram is a rectangle or a rhombus.
To find the length of the diagonals of a parallelogram, we can use the properties of parallelograms.
1. Start by drawing a parallelogram with the given dimensions, labeling the angles and sides as necessary.
10 cm
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6 cm | / ? cm (diagonal)
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2. Remember that opposite angles in a parallelogram are congruent. Since one angle is 70 degrees, the opposite angle is also 70 degrees.
3. Look at the parallelogram. Notice that the two shorter sides (6 cm) are the adjacent sides to the 70-degree angle.
4. We can use the Law of Sines to find the length of the diagonal.
The Law of Sines states that for any triangle:
sin(A) / a = sin(B) / b = sin(C) / c
In this case, the triangle formed by the diagonal, the 70-degree angle, and one of the shorter sides (6 cm) can be used to find the length of the diagonal.
Let's call the length of the diagonal 'd'.
So, sin(70 degrees) / 6 cm = sin(90 degrees) / d
Since sin(90 degrees) equals 1, the equation simplifies to:
sin(70 degrees) = 6 cm / d
5. To find sin(70 degrees), we can use a scientific calculator or online calculator.
sin(70 degrees) ≈ 0.9397
6. Substitute the values in the equation:
0.9397 = 6 cm / d
7. Solve for d:
d = 6 cm / 0.9397
8. Using a calculator, calculate:
d ≈ 6.39 cm
Therefore, the length of the diagonals of the parallelogram is approximately 6.39 cm.