A rhombus has diagonal of length 12 cm and 7 cm respectively.find the larger angle of the rhombus. How do you find the larger angle of the rhombus?
the diagonals are perpendicular bisectors of each other
so the tangent of half of the angle is ... 6 / 3.5
To find the larger angle of a rhombus, we can use the properties of rhombus. Here's how you can find the larger angle:
Step 1: Recall that the diagonals of a rhombus bisect each other at right angles.
Step 2: Draw a rhombus and label the diagonals as AC and BD, where AC is the longer diagonal. Let O be the point of intersection of the diagonals.
Step 3: Since the diagonals of a rhombus bisect each other, AO = OC and BO = OD.
Step 4: Since each diagonal of a rhombus divides the rhombus into two congruent triangles, angle AOB and angle BOD are congruent.
Step 5: Therefore, we can calculate the measure of one of these angles by using the properties of a right-angled triangle. In triangle AOB, angle AOB is a right angle, and we have the length of both sides AO (half of the longer diagonal) and OB (half of the shorter diagonal).
Step 6: To find angle AOB, we can use the trigonometric relationship of tangent. Since tangent is the ratio of the opposite side to the adjacent side, we have:
tan(AOB) = AO / OB
Step 7: Substitute the values into the equation. The value of AO is half of the longer diagonal (AC) and the value of OB is half of the shorter diagonal (BD).
tan(AOB) = (1/2 * AC) / (1/2 * BD)
Step 8: Simplify the equation by canceling out the common 1/2 term:
tan(AOB) = AC / BD
Step 9: Now we can calculate angle AOB by taking the inverse tangent (arctan) of both sides of the equation:
AOB = arctan(AC / BD)
Step 10: Substitute the given values into the equation. AC is 12 cm (the longer diagonal) and BD is 7 cm (the shorter diagonal).
AOB = arctan(12 / 7)
Step 11: Use a calculator to find the value of arctan(12 / 7).
AOB ≈ 58.43°
Step 12: Since angle BOD is congruent to angle AOB, the larger angle of the rhombus is 2 times the value of angle AOB.
Larger angle = 2 * AOB ≈ 2 * 58.43°
Larger angle ≈ 116.86°
Therefore, the larger angle of the rhombus is approximately 116.86 degrees.
To find the larger angle of a rhombus, you can use the formula:
Larger angle = 180 - Smaller angle
Here's how you can calculate the smaller angle of the rhombus using the given diagonal lengths:
Step 1: Draw the rhombus and label the diagonals. Let's call the length of the first diagonal "d1" and the length of the second diagonal "d2". In this case, d1 = 12 cm and d2 = 7 cm.
Step 2: Notice that the diagonals of a rhombus divide it into four congruent right-angled triangles.
Step 3: Use the Pythagorean theorem to find the length of one side of each triangle. The Pythagorean theorem states that for a right-angled triangle, the sum of the squares of the two legs (sides that form the right angle) is equal to the square of the hypotenuse (the side opposite the right angle).
In this case, we have:
Side length^2 + Side length^2 = Hypotenuse^2
Let's call the side length "s".
s^2 + s^2 = (d1/2)^2
s^2 + s^2 = 6^2 (since d1 = 12 cm, half of d1 is 6 cm)
2s^2 = 36
s^2 = 18
s = sqrt(18) (taking the square root on both sides)
Step 4: Now that we know the length of one side of the rhombus (sqrt(18) cm), we can find the smaller angle.
To find the smaller angle, we can use any of the four triangles formed by the diagonals. Since all the triangles are congruent, the angles will be the same.
In a right-angled triangle, the two acute angles add up to 90 degrees. So, each acute angle in the triangle will be 45 degrees.
Therefore, the smaller angle of the rhombus is 45 degrees.
Step 5: Finally, we can find the larger angle of the rhombus using the formula:
Larger angle = 180 - Smaller angle
Larger angle = 180 - 45
Larger angle = 135 degrees
Thus, the larger angle of the rhombus is 135 degrees.