Problem of the Week
Alternate Dimensions
The four shapes to the right are each drawn
with a horizontal base and a vertical height.
Figure A is a right-angled triangle, Figure B
is an isosceles triangle, Figure C is a square,
and Figure D is a rectangle. The figures are
not drawn to scale.
Using the following clues, determine the measure of the (horizontal) base and the
measure of the (vertical) height of each figure.
1. The measure of the base of Figure A is the same as the measure of the base
of Figure D.
2. The measure of the base of Figure A is one unit less than the measure of the
base of Figure B.
3. The side length of Figure C is the same as the measure of the base of
Figure A.
4. The measure of the height of Figure B is the same as the measure of the
height of Figure A and also the same as the measure of the base of Figure B.
5. The area of Figure C is 9 square units.
6. The total area of all four figures is 38 square units.
A = the total area of all four figures
AA = the area of a right-angled triangle
AB = the area of a an isosceles triangle
AC = the area of a an square
AD = the area of a rectangle
LA = the measure of the base of Figure A
LB = the measure of the base of Figure B
LC = the measure of the base of Figure C
LD = the measure of the base of Figure D
LC = the side length of Figure C
hA = measure of the height of Figure A
hB =measure of the height of Figure B
hD = measure of the height of Figure D
Clues:
1.
The measure of the base of Figure A is the same as the measure of the base of Figure D mean:
LA = LD
2.
The measure of the base of Figure A is one unit less than the measure of the base of Figure B mean:
LA = LB - 1
3.
The side length of Figure C is the same as the measure of the base of Figure A mean:
LC = LA
4.
The measure of the height of Figure B is the same as the measure of the height of Figure A and also the same as the measure of the base of Figure B mean:
hB = hA = LB
5.
5. The area of Figure C is 9 square units mean:
LC² = 9
So:
LC² = √9
LC = 3
LC = LA
LA = LC
LA = 3
LA = LB - 1
3 = LB - 1
Add 1 to both sides
4 = LB
LB = 4
LA = LD
LD = LA
LD = 3
hB = hA = LB
hA= LB
hA = 4
hB = LB
hB = 4
6.
AA = area of a right-angled triangle
AB = area of a an isosceles triangle
AC = area of a an square
AD = area of a rectangle
A = AA + AB + AC + AD = 38
AA = LA ∙ hA / 2
AA = 3 ∙ 4 / 2 = 12 / 2
AA = 6
AB = LB ∙ hB / 2
AB = 4 ∙ 4 / 2 = 16 / 2
AB = 8
AC = 9
AD = LD ∙ hD
AD = 3 ∙ hD
The total area of all four figures:
A = area of a right-angled triangle + area of a an isosceles triangle + area of a an square + area of a rectangle = 38
A = AA + AB + AC + AD = 38
6 + 8 + 9 + 3 hD = 38
23 + 3 hD = 38
Subtract 23 to both sides
3 hD = 15
Divide both sides by 3
hD = 5
AD = LD ∙ hD
AD = 3 ∙ 5
AD = 15
Results:
AA = the area of a right-angled triangle = 6
AB = the area of a an isosceles triangle = 8
AC = the area of a an square = 9
AD = the area of a rectangle = 15
LA = the measure of the base of Figure A = 3
LB = the measure of the base of Figure B = 4
LC = the side length of Figure C = 3
LD = the measure of the base of Figure D = 3
hA = measure of the height of Figure A = 4
hB =measure of the height of Figure B = 4
hD = measure of the height of Figure D = 5
One my typo:
It should not be written like this:
So:
LC² = √9
It should be written like this:
So:
LC = √9
To solve this problem, we will use the given clues to determine the measurements of the bases and heights of the four figures. Let's go through each clue and use them to find the solutions.
1. The measure of the base of Figure A is the same as the measure of the base
of Figure D.
This clue tells us that the base of Figure A is equal to the base of Figure D. However, we don't know the exact measurement yet, so let's represent it as 'x' for now.
2. The measure of the base of Figure A is one unit less than the measure of the
base of Figure B.
According to this clue, the base of Figure A is one unit less than the base of Figure B. So, we can write the equation:
Base of Figure A = Base of Figure B - 1
Since we don't know the exact measurement of the base of Figure B yet, we'll represent it as 'y' for now.
3. The side length of Figure C is the same as the measure of the base of
Figure A.
This clue tells us that the side length of Figure C is equal to the base of Figure A. We can represent this as:
Side length of Figure C = Base of Figure A = x
4. The measure of the height of Figure B is the same as the measure of the
height of Figure A and also the same as the measure of the base of Figure B.
According to this clue, the height of Figure B is the same as the height of Figure A, which is also equal to the base of Figure B. Let's represent the height of Figure B as 'h'.
Height of Figure B = Height of Figure A = Base of Figure B = h
5. The area of Figure C is 9 square units.
This clue gives us the area of Figure C, which is 9 square units. Since Figure C is a square, we know that the area is calculated by multiplying the length of one side by itself. So:
Area of Figure C = Side length of Figure C * Side length of Figure C
= x * x
= x^2
Given that x^2 = 9, we can find the value of x by taking the square root of both sides:
√(x^2) = √9
x = 3
Now, we have the value of x, which is 3.
6. The total area of all four figures is 38 square units.
This clue tells us that the total area of all four figures combined is 38 square units. We can calculate the area of each figure and add them to get the total.
Area of Figure A = (Base of Figure A * Height of Figure A) = (x * h)
Area of Figure B = (Base of Figure B * Height of Figure B) = (h * h) = h^2
Area of Figure C = (Side length of Figure C * Side length of Figure C) = (x * x) = x^2
Area of Figure D = (Base of Figure D * Height of Figure D) = (x * h)
Total Area = Area of Figure A + Area of Figure B + Area of Figure C + Area of Figure D
Total Area = xh + h^2 + x^2 + xh
Since the total area is given as 38 square units, we can write the equation:
xh + h^2 + x^2 + xh = 38
Now let's substitute the values we know:
Using x = 3 from clue 5, we get:
3h + h^2 + 9 + 3h = 38
Combining like terms:
2h + h^2 = 29
At this point, we don't have enough information to find the exact values of h or the bases of the figures. However, we can use trial and error to check for valid solutions that satisfy all the clues and the total area of 38 square units.