Question

select the correct answer from the drop - down menu.
a university just acquired more land which is shown by the triangle in the diagram. what is the area of the land that the university just acquired?
the area is
square miles.
(diagram description: a triangle with university road (2 miles, perpendicular to the base) as height, and the base is composed of two segments each 1.5 miles? or the base length can be inferred from the diagram. the sides bike trail and oak street are 2.5 miles each.)

Explanation:

Step1: Identify the base and height

The triangle is isoceles? Wait, no, looking at the diagram, the base can be found by adding the two segments on either side of University Road. Wait, the right triangle on the left has base 1.5 miles (since Oak Street to University Road is 1.5? Wait, no, the diagram shows that the perpendicular from the vertex to the base (Euclid Avenue) is split into two parts? Wait, no, actually, the triangle is a triangle with base length equal to the sum of the two segments? Wait, no, looking at the diagram, the two right triangles (left and right) are congruent? Wait, the left triangle has base 1.5 miles (since the horizontal segment from Oak Street to University Road is 1.5), and the right triangle should also have base 1.5 miles? Wait, no, the total base of the big triangle (Euclid Avenue) is 1.5 + 1.5 = 3 miles? Wait, no, wait the height is 2 miles? Wait, no, the University Road is 2 miles? Wait, the diagram: the vertical segment (University Road) is 2 miles, and the horizontal segments (from University Road to Oak Street and to bike trail) are 1.5 miles each? Wait, no, the left triangle: Oak Street to University Road is 1.5 miles (horizontal), height (vertical) is 2 miles? Wait, no, the length of Oak Street is 2.5 miles, which is the hypotenuse of the left right triangle. Let's check: 1.5² + 2² = 2.25 + 4 = 6.25, and 2.5² = 6.25. So that's a right triangle. So the left right triangle has legs 1.5 and 2, hypotenuse 2.5. Similarly, the right right triangle (bike trail side) should also have legs 1.5 and 2, hypotenuse 2.5. So the total base of the big triangle (Euclid Avenue) is 1.5 + 1.5 = 3 miles, and the height (the vertical segment, University Road) is 2 miles? Wait, no, the height is the perpendicular distance from the top vertex to the base (Euclid Avenue), which is 2 miles? Wait, no, the University Road is 2 miles, but is that the height? Wait, the diagram shows that the perpendicular from the top vertex to Euclid Avenue is University Road, which is 2 miles. Then the base of the triangle is the length of Euclid Avenue. Since the two right triangles (left and right) are congruent (both have hypotenuse 2.5, one leg 2, so the other leg is 1.5), so the base is 1.5 + 1.5 = 3 miles. Then the height is 2 miles? Wait, no, wait the height is the length of the perpendicular, which is 2 miles? Wait, no, the University Road is 2 miles, but is that the height? Wait, the area of a triangle is (base height)/2. So base is 3 miles (1.5 + 1.5), height is 2 miles? Wait, no, wait the left triangle: base 1.5, height 2, area (1.52)/2 = 1.5. The right triangle: same, area 1.5. So total area is 1.5 + 1.5 = 3? Wait, no, that can't be. Wait, maybe the height is 2 miles, and the base is 3 miles? Wait, (3 2)/2 = 3. Wait, but let's recheck. The left right triangle: legs 1.5 and 2, area (1.52)/2 = 1.5. The right right triangle: same, area 1.5. So total area is 3. Alternatively, the big triangle has base 3 (1.5+1.5) and height 2, so area (3*2)/2 = 3.

Step2: Calculate the area

Using the formula for the area of a triangle: \( A = \frac{1}{2} \times \text{base} \times \text{height} \). The base is \( 1.5 + 1.5 = 3 \) miles, and the height is 2 miles. So \( A = \frac{1}{2} \times 3 \times 2 \).

Step3: Compute the area

\( \frac{1}{2} \times 3 \times 2 = 3 \). Wait, but let's check again. Wait, maybe the height is 2 miles, and the base is 3 miles? Wait, the left triangle: base 1.5, height 2, area 1.5. Right triangle: base 1.5, height 2, area 1.5. Total 3. So the area is 3 square miles.

Answer:

3