the lengths of the diagonal of a rhombus are 2 in and 5 in.
find the measure of the angles of the rhombus to the nearest degree.
if it helps, we are learning how to use tangent right now!
Draw a rhombus with the diagonals.
4 right triangles are formed.
The intersection of the diagonals of a rhombus form 90 degree (right) angles (property of a rhombus).
The diagonals of a rhombus bisect each other. This means that they cut each other in half.
So, each of the right triangles have one side = 1, one side = 2.5 and the hypotenuse = unknown (sides of the
rhombus).
Triangle 1. ABC
side a = 2.5, side b = 1
tan A = a/b = 2.5/1 = 2.5
tan A = 2.5
tan 68.2 deg = 2.5
Since, angle C = 90 and angle A = 68.2
angle B = 21.8 deg. (180 - (90 + 68.2))
Since, the diagonals bisect angles in each corner of a rhombus (property of a rhombus), therefore, one corner angle is 68.2 + 68.2 = 136.4 deg. and the same side other corner angle is 21.8 + 21.8 = 43.6 deg.
Adjacent sides (ones next to each other) of a rhombus are supplementary. This means that their measures add up to 180 degrees.
Therefore, the opposite corners are
180 - 136.4 = 43.6 and
180 - 43.6 = 136.4.
So, the four angles are,
136.4, 43.6, 136.4, and 43.6
Well, isn't it fabulous that your rhombus has diagonals of 2 inches and 5 inches! Get ready for some diagonal fun with tangent!
First, let's use the properties of a rhombus to our advantage. We know that the diagonals of a rhombus are always perpendicular bisectors of each other. So, we can draw a right triangle using one of the diagonals and its midpoint. Let's call the shorter diagonal "a" (2 inches) and the longer one "b" (5 inches).
Using the mighty Pythagorean theorem, we can find the length of the side of the rhombus (let's call it "s"). So, s^2 = (a/2)^2 + (b/2)^2. Simplifying, we get s^2 = 1^2 + 2.5^2. Calculating further, s^2 = 7.25. Taking the square root of both sides, we find s ≈ 2.69 inches.
Now that we know the length of one side, we can calculate the measures of the angles using tangent. Since the diagonals of a rhombus bisect the angles, let's focus on just one of the four angles (let's call it θ).
Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle. In our case, the opposite side is half the length of one of the diagonals (represented by "a/2") and the adjacent side is half the length of the other diagonal (represented by "b/2").
Using the equation tan(θ) = (a/2)/(b/2), we get tan(θ) = 1/2. Taking the arctan of both sides, we find θ ≈ 26.57 degrees.
Since opposite angles of a rhombus are congruent, the other three angles will also be approximately 26.57 degrees.
So, to the nearest degree, the measure of each angle in the rhombus is 27 degrees.
Hope this tangled tangent tale helped!
To find the measure of the angles of a rhombus, we can use the properties of a rhombus.
1. In a rhombus, opposite angles are congruent, which means they have the same measure.
2. The diagonals of a rhombus bisect each other at 90 degrees.
Knowing these properties, we can find the measure of the angles step-by-step:
Step 1: Draw the rhombus and label the given lengths of the diagonals. Let's call the shorter diagonal "d1" and the longer diagonal "d2."
We have d1 = 2 in and d2 = 5 in.
Step 2: Divide each diagonal by 2 to find the lengths of the half-diagonals.
Half of the shorter diagonal (d1/2) = 2/2 = 1 in.
Half of the longer diagonal (d2/2) = 5/2 = 2.5 in.
Step 3: Draw the half-diagonals from the intersection point of the diagonals to the opposite vertices of the rhombus.
This divides the rhombus into four congruent right-angled triangles.
Step 4: Use the tangent function to find the angle measure.
Tangent (θ) = Opposite / Adjacent
In each right-angled triangle, the opposite side is half of one diagonal and the adjacent side is half of the other diagonal.
So, for each triangle, we can find the value of θ using the tangent function.
For the triangle with a half-diagonal of 1 inch:
Tangent (θ) = Opposite side / Adjacent side
Tangent (θ) = 1 in / 2.5 in
θ = arctan(1/2.5) ≈ 21.8 degrees
Since opposite angles in a rhombus are congruent, the measure of each angle of the rhombus is twice θ.
Therefore, each angle of the rhombus measures 2 * 21.8 ≈ 43.6 degrees.
So, the measure of each angle of the rhombus, to the nearest degree, is approximately 44 degrees.
To find the measure of the angles of a rhombus, we can use trigonometry and the given lengths of the diagonals.
First, let's consider the properties of a rhombus. A rhombus is a quadrilateral with four congruent sides and opposite angles that are equal.
We are given that the lengths of the diagonals are 2 inches and 5 inches. Let's denote the lengths of the diagonals as d1 and d2.
Now, let's use the properties of a rhombus to find the measure of the angles.
1. Draw the rhombus and label the diagonals d1 and d2.
---- d1 ----
| |
| |
d2 | | d2
| |
| |
---- d1 ----
2. Since the diagonals of a rhombus bisect each other perpendicularly, we can split the rhombus into four congruent right triangles.
3. For each right triangle, we can use the tangent function to find the measure of the acute angle opposite to d1.
tan(angle) = opposite / adjacent
In this case, the opposite side is d2/2 (since it is half the length of d2), and the adjacent side is d1/2 (since it is half the length of d1).
tan(angle) = (d2/2) / (d1/2)
tan(angle) = d2 / d1
4. Now, we can find the tangent of the angle by dividing the length of d2 by the length of d1.
tan(angle) = 5 / 2
angle = atan(5/2)
Using a calculator, the angle is approximately 67.38 degrees.
5. Since adjacent angles of a rhombus are supplementary (sum of two angles is 180 degrees), we can find the adjacent angle by subtracting the obtained angle from 180 degrees.
adjacent angle = 180° - 67.38°
adjacent angle = 112.62°
6. Finally, since the opposite angles of a rhombus are equal, we have found two opposite angles. To find the other two angles, we subtract the adjacent angle from 180 degrees.
other angle = 180° - 112.62°
other angle = 67.38°
Therefore, the measure of each angle in the rhombus, to the nearest degree, is 67°.