Question

13 explain two ways you can tell which triangles in problems 1–6 have an area greater than 15 square units.

Explanation:

Way 1: Using the Area Formula

The area of a triangle is given by the formula \( A=\frac{1}{2}bh \), where \( b \) is the base and \( h \) is the height (the perpendicular distance from the base to the opposite vertex). For each triangle in problems 1 - 6, we can identify the base and height (either from the given diagram or problem description). Then we substitute these values into the formula to calculate the area. If the calculated area \( A > 15 \) square units, then the triangle has an area greater than 15 square units. For example, if a triangle has a base \( b = 10 \) units and a height \( h = 4 \) units, then \( A=\frac{1}{2}\times10\times4 = 20 \) square units, which is greater than 15.

Way 2: Comparing with a Reference Triangle

We can create a reference triangle with an area of 15 square units. Using the area formula \( A=\frac{1}{2}bh = 15 \), we can find possible combinations of base and height, such as \( b = 6 \) and \( h = 5 \) (since \( \frac{1}{2}\times6\times5=15 \)), or \( b = 5 \) and \( h = 6 \), or \( b = 10 \) and \( h = 3 \) ( \( \frac{1}{2}\times10\times3 = 15 \) ), etc. Then, for each triangle in problems 1 - 6, we can visually compare (if diagrams are available) or compare the base - height product (since \( A=\frac{1}{2}bh \), the product \( bh \) is proportional to the area) with the product of the base and height of our reference triangle. If the product \( bh \) of the triangle in question is greater than \( 30 \) (because \( \frac{1}{2}bh>15\Rightarrow bh > 30 \)), then its area is greater than 15 square units. For example, if a triangle has \( b = 7 \) and \( h = 5 \), then \( bh=35>30 \), so \( A=\frac{1}{2}\times35 = 17.5>15 \).

Answer:

Two ways are: 1) Use the triangle area formula \( A = \frac{1}{2}bh \) to calculate the area of each triangle and check if it is greater than 15. 2) Compare the base - height product (\( bh \)) of each triangle with 30 (since \( \frac{1}{2}bh>15\Rightarrow bh > 30 \)) or compare with a reference triangle of area 15.