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.
there are three quantities: area, base and height.
Knowing any two will allow you to find the third.
Let's solve this step by step:
Step 1: Determine the measure of the base of Figure A and Figure D.
Based on Clue 1, the measure of the base of Figure A is the same as the measure of the base of Figure D.
Step 2: Determine the measure of the base of Figure B.
Based on Clue 2, the measure of the base of Figure A is one unit less than the measure of the base of Figure B.
Step 3: Determine the measure of the side length of Figure C.
Based on Clue 3, the side length of Figure C is the same as the measure of the base of Figure A.
Step 4: Determine the measure of the height of Figure B.
Based on Clue 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.
Step 5: Determine the area of Figure C.
Based on Clue 5, the area of Figure C is 9 square units.
Step 6: Determine the total area of all four figures.
Based on Clue 6, the total area of all four figures is 38 square units.
Now let's solve it:
Step 1: Since the measure of the base of Figure A is the same as the measure of the base of Figure D, let's say the base of Figure A and D is 'x' units.
Step 2: According to Clue 2, the measure of the base of Figure A is one unit less than the measure of the base of Figure B. So, the base of Figure B is 'x + 1' units.
Step 3: According to Clue 3, the side length of Figure C is the same as the measure of the base of Figure A. So, the side length of Figure C is 'x' units.
Step 4: According to Clue 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. Therefore, the height of Figure B is 'x + 1' units.
Step 5: According to Clue 5, the area of Figure C is 9 square units. Since Figure C is a square, the area is calculated by multiplying the side length by itself. So, 'x * x = 9'. Solving this equation gives us 'x = 3'.
Step 6: According to Clue 6, the total area of all four figures is 38 square units. Let's calculate the areas of each figure and add them up.
- The area of Figure A (right-angled triangle) is calculated by multiplying the base and the height and dividing by 2. So, the area of Figure A is '(x * height of A) / 2 = (3 * (x + 1)) / 2'.
- The area of Figure B (isosceles triangle) is calculated by multiplying the base and the height and dividing by 2. So, the area of Figure B is '((x + 1) * (x + 1)) / 2 = (x + 1)^2 / 2'.
- The area of Figure C (square) is given as 9 square units. So, the area of Figure C is 9.
- The area of Figure D (rectangle) is calculated by multiplying the base and the height. So, the area of Figure D is 'x * height of D = 3 * height of D'.
Now, we can set up an equation to find the height of Figure A and Figure D:
(3 * (x + 1)) / 2 + (x + 1)^2 / 2 + 9 +3 * height of D = 38
Solving this equation will give us the height of Figure A and Figure D.
Let's calculate the final values:
- x = 3
- The base of Figure A and Figure D = x = 3 units
- The base of Figure B = x + 1 = 4 units
- The side length of Figure C = x = 3 units
- The height of Figure B = x + 1 = 4 units
- The height of Figure A and Figure D = Solving the equation: (3 * (3+1)) / 2 + (3+1)^2 / 2 + 9 + 3 * height of D = 38, gives us the height of Figure A and Figure D = 6 units.
To determine the measures of the base and height of each figure, let's analyze the clues given and solve the problem step by step.
1. The measure of the base of Figure A is the same as the measure of the base of Figure D.
Let's denote the base of Figure A as 'x'. Therefore, the base of Figure D is also 'x'.
2. The measure of the base of Figure A is one unit less than the measure of the base of Figure B.
Let's denote the base of Figure B as 'y'. According to the clue, x = y - 1.
3. The side length of Figure C is the same as the measure of the base of Figure A.
Given that Figure C is a square, the base and height of Figure C are equal to the side length. Therefore, the side length of Figure C is also '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.
Let's denote the height of Figure A and Figure B as 'h'. Therefore, h = y = base of Figure B.
5. The area of Figure C is 9 square units.
Since Figure C is a square, its area is given by the formula: area = side length * side length. From clue 3, we know that the side length is 'x'. Hence, x * x = 9. This gives us x = √9 = 3.
6. The total area of all four figures is 38 square units.
We can calculate the areas of Figures A, B, and D using the formula: area = base * height.
- The area of Figure A = x * h = 3 * h
- The area of Figure B = y * h = (x + 1) * h = (3 + 1) * h = 4h
- The area of Figure D = x * h = 3h
To find the value of 'h', we need to determine the sum of the areas of Figures A, B, C, and D and equate it to 38.
area(A) + area(B) + area(C) + area(D) = 38
3h + 4h + 9 + 3h = 38
10h + 9 = 38
10h = 38 - 9
10h = 29
h = 29/10 = 2.9
Finally, we have the following measures for the base and height of each figure:
- Figure A: base = 3, height = 2.9
- Figure B: base = 4, height = 2.9
- Figure C: base = 3, height = 3 (since it is a square)
- Figure D: base = 3, height = 2.9