Question

Use a straightedge to draw the following figures. a. convex pentagon with two perpendicular sides b. concave hexagon with three consecutive congruent sides

Short answer

Draw a pentagon with two right-angle sides, and a hexagon with three equal-length consecutive sides and one angle > 180°.

Step-by-step solution

Define the Properties of the Convex Pentagon

A convex pentagon is a five-sided figure where all interior angles are less than 180 degrees. You need to include two sides that intersect at a right angle (90 degrees) to meet the condition of perpendicularity.

Draw the Perpendicular Sides

Using a straightedge, draw two lines intersecting at one point with a 90-degree angle between them. This will be the starting point for your convex pentagon with perpendicular sides.

Complete the Pentagon

From the endpoints of the perpendicular sides, draw three more lines such that they connect back to form a pentagon. Ensure that the overall shape remains convex, without any interior angles exceeding 180 degrees.

Define the Properties of the Concave Hexagon

A concave hexagon has at least one interior angle greater than 180 degrees. For this problem, three of its sides must be of equal length, and they must be consecutive, meaning they follow each other in sequence.

Draw the Three Congruent Sides

Start by drawing three consecutive equal-length sides using the straightedge to ensure they are straight and equal. These will form the basis for the hexagon.

Complete the Hexagon Ensuring Concavity

Add three more sides to complete the hexagon, ensuring that at least one interior angle exceeds 180 degrees, which creates a concavity in the shape. Ensure the shape closes back to your starting point.

Key concepts

Convex Pentagon

A convex pentagon is a geometric figure with five sides, where each internal angle is less than 180 degrees. Convex shapes have no indentations or inward curves. This means the entire shape bulges outward. When constructing such a shape, two sides need to intersect at a 90-degree angle to satisfy the requirement of perpendicularity. A perpendicular intersection implies that two lines meet to form a right angle. This is crucial in retaining the convex nature while forming a pentagon.

• Begin by drawing two lines that intersect at a right angle.
• These right-angled sides will form the "corner" of your pentagon.
• From the ends of these sides, draw three more lines to complete the shape.

Make sure these entire five edges connect to maintain a convex outline, which means verifying each interior angle remains under 180 degrees.

Concave Hexagon

A concave hexagon is distinct in that it includes at least one interior angle greater than 180 degrees, creating an indentation or "cave." In this exercise, it's important that three sides are consecutively congruent. These sides should be equal in length and directly follow one another. To draw this figure, begin with the congruent sides as these will form a foundation for structuring the hexagon.

• Use a straightedge to ensure these three sides are of equal length.
• Arrange them consecutively, meaning one after another, forming a part of the hexagon's border.

Next, add additional sides to ensure the overall structure closes, while paying attention to the internal angles. At least one angle should exceed 180 degrees to achieve the concave characteristic. This distinctive angle is where the "cave" appears, making the hexagon interesting and satisfying the exercise's requirement.

Perpendicular Sides

Perpendicular sides in geometry refer to two lines forming a 90-degree angle where they intersect. This is commonly seen in right-angled triangles or as part of the conditions defined for certain polygons, like the pentagon in the exercise. When constructing these sides on paper with a straightedge, it is vital to ensure the correct degree measure for precision and accuracy.

• Perpendicular lines create a distinct "L" shape, making them easy to identify.
• Geometrically, they provide stability and are often the basis for many standard geometric designs.

To ensure correct perpendicularity, consider using a set square or right angle drawing tool. These tools help in maintaining the 90-degree intersection, crucial for meeting mathematical and aesthetic requirements in geometric artwork.