You are bending a wire to make a coat hanger. The length of the wire is 88 centimeters, and 20 centimeters are needed to make a hook hanger. The triangular portion of the hanger is an isosceles triangle. The length of one leg of this triangle is 3/5 the length of the base. Sketch the hanger. Give the dimensions of the triangular portion.
The triangle has a base of length b, and two legs of 0.6b each.
b + 2*.6b + 20 = 88
2.2b = 68
b = 30.9 cm
3/5 b = 18.55 cm
30.9 + 2*18.55 + 20 = 88.0
To sketch the hanger and provide the dimensions of the triangular portion, we'll break it down into steps:
Step 1: Start by sketching a straight line to represent the wire. Label the length of the wire as 88 centimeters.
Step 2: Next, draw a vertical line perpendicular to the wire line on one end to create the hook portion. Label this vertical line as 20 centimeters.
Step 3: Now, we need to construct the triangular portion of the hanger. To do this, measure and mark a point on the wire line such that it is 3/5 the length of the base (wire line). This will be the midpoint of the base of the isosceles triangle.
Step 4: From this midpoint, draw two diagonal lines upwards, connecting with the upper portion of the vertical line (hook) you created in step 2. These diagonal lines will form the legs of the isosceles triangle.
Step 5: Finally, label the base of the isosceles triangle as the wire length between the start point and the midpoint (found in step 3). Label the length of the leg(s) of the triangle if needed.
The resulting sketch should accurately depict the coat hanger, with the dimensions of the triangular portion as specified. Keep in mind that the exact measurements of the triangular portion cannot be determined without more information about the specific proportions (angles, ratio of leg lengths, etc.)
To find the dimensions of the triangular portion of the hanger, we need to determine the lengths of the base and the legs of the isosceles triangle.
Let's start by assigning variables to the lengths of the base (B) and the legs (L) of the triangle.
Given:
Total length of wire (Lw) = 88 cm
Length needed to make the hook (Lh) = 20 cm
Length of one leg of the triangle (L) = 3/5 times the length of the base (B)
We can set up the following equation to represent the lengths of the wire:
Lw = L + L + B + Lh
Substituting the values we have:
88 cm = L + L + B + 20 cm
Simplifying the equation:
88 cm - 20 cm = 2L + B
68 cm = 2L + B
Now, we can use the relationship between the base and the leg given in the problem:
L = (3/5)B
Substituting this into the previous equation:
68 cm = 2(3/5)B + B
Simplifying further:
68 cm = (6/5)B + B
Multiplying both sides by 5 to get rid of the fraction:
340 cm = 6B + 5B
Combining like terms:
340 cm = 11B
Now, we can solve for B, the length of the base:
B = 340 cm / 11
B ≈ 30.91 cm
To find the length of the leg:
L = (3/5)B
Substituting the value of B:
L = (3/5)(30.91 cm)
L ≈ 18.55 cm
Therefore, the base of the triangular portion of the hanger is approximately 30.91 cm and the legs are approximately 18.55 cm each.
Sketch of the hanger:
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