Question

Need steps how to solve​

Answer 1

Answer:

1. Perimeter: 164 cm

2. x = 21°; y=146°

3. x = 112°; y = 112°

4.  x = 72°; y = 61°

Step-by-step explanation:

Your figures are all kites. We know that because a kite is a quadrilateral that has 2 pairs of adjacent sides that have the same length and 1 pair of congruent angles. Look at the picture attached, so you can see it.

1. So for figure one to get the perimeter, you just add all the sides together. But as you can, you need to solve for x first so you can solve for the measure of the longer sides.

The two longer sides are adjacent to each other and they are one of the pairs that are congruent. So this means that:

x + 35 = 2x - 5

So let's solve for x first:

x + 35 = 2x - 5

Add 5 on both sides of the equation:

x + 35 + 5= 2x - 5 + 5

x + 40 = 2x

Subtract x on both sides of the equation:

x - x + 40 = 2x - x

40 = x

Now that we solved that, let's get the measure of the longer sides. You just need to solve for one because they are congruent:

x + 35

40 + 35 = 75cm

Now let's go to the shorter pair. They are also one of the two adjacent, congruent sides, so the missing side is 7cm.

Now let's add all the sides together:

75 cm + 75 cm + 7 cm + 7 cm = 150 cm + 14 cm = 164 cm

2. x = _______ y = ________

Like we mentioned earlier, a kite has 2 congruent angles. The angles that are congruent are the angles in between the short side and the long side, these are called non-vertex sides. So in this case, 146° is the measure of one of the angles, and it is congruent to the angle across it.

And so:

y = 146°

Also earlier, we mentioned that a kite is a quadrilateral. The sum of the angles of a quadrilateral is = 360°. We can then assume that:

x + y + 146° + 47° = 360°

So we plug in the value of y and solve for x:

x + 146° + 146° + 47° = 360°

x + 339° = 360°   (Subtract  339° on both sides of the equation)

x + 339° - 339° = 360° - 339°

x = 21 °

3. Using the same reasoning as the previous problem, we know that all the angles sum up to 360°.

x + y + 101° + 35° = 360°

Notice that x and y are non-vertex angles, and non-vertex angles of a kite are congruent, so x = y. Now we rewrite the equation knowing that:

x + y + 101° + 35° = 360°

x + x + 101° + 35° = 360°

2x + 101° + 35° = 360°

2x + 136° = 360°

2x = 360° - 136°

2x/2 = 224°/2

x = 112°

4. Earlier I mentioned non-vertex angles of the kite. Now the other two angles are called vertex angles. The long diagonal (The longer line that goes across the kite), bisects the vertex angles, this means that they  separate the angle evenly into two.

Now notice that the short diagonals divide the kite into 2 isosceles triangles. Isosceles triangles are triangles with 2 congruent legs and the two angles the legs make with the base are congruent. Now all the angles in a triangle sum up to 180°.

Let's take the triangle that the longer sides make.

Look at image P4 attached, we're working on the red triangle:

Red triangle:

We solve for the angle across the base:

29° + 29° = 58°

Now because we know that the two legs are equal the angles they make are equal too.

y + y + 58° = 180°

2y + 58° = 180°

2y = 180° - 58°

2y/2 = 122°/2

y = 61°

Now let's solve for the angles of the blue triangle of P4

We can solve for x by first solving for the angle as a whole. Since the long diagonal bisects it, that means that the whole angle is 2x.

The legs are also equal so that means that this is also an isosceles angle. Now we solve this knowing that the angles that the legs and the base make are congruent.

18° + 18° + 2x = 180°

36° + 2x = 180°

2x = 180° - 36°

2x = 144°

x = 72°