Tide-Pool Survey: Design the Boundaries

Construct survey shapes and decide which conditions fix their size and shape.

Plan the Survey Boundaries

A marine biology class is planning outlines for counting tide-pool species from photographs. All lengths are measurements on the photographs, not animal counts. Rotations and mirror images count as the same shape.

To outline a snail-counting zone, students draw a base and two rays above it. The interior angle at the left endpoint is 65°, and the interior angle at the right endpoint is 115°. What will happen?

A. The rays will meet to form one triangle.B. The rays will be parallel, so no triangle forms.C. The rays will meet to form two different triangles.D. The rays will allow triangles of many sizes.

A triangular crab-counting zone must have two equal interior angles. Its third interior angle is 44°.

What must each of the equal angles measure? °

The class wants a right-triangular mussel-counting zone. Each record lists all three proposed interior angles. Select every record that can describe a right triangle.

Sam and Imani each construct a triangular survey outline with sides of 4.5 cm, 6 cm, and 8 cm. Sam's triangle points left, and Imani's points right. What do the measurements determine?

A. They determine one size and shape, despite the different directions.B. They determine many sizes and shapes because no angles are given.C. They determine two sizes and shapes because the directions differ.D. They determine no triangle because the longest side is too long.

Construct triangle PQR for a sea-star counting zone. Draw PQ = 5.5 cm horizontally, with P on the left. Above PQ, construct an interior angle of 40° at P and an interior angle of 65° at Q. Label the intersection of the rays R. Draw the full triangle and label the given measurements.

Test the Measurements

Use the side and angle conditions to check each proposed outline. A triangle must have three corners; a straight line does not count.

A triangular limpet-counting outline has a perimeter of 24 cm. Two sides measure 7 cm and 8 cm.

What length must the third side have? cm

Two sides of a triangular anemone-counting outline are 3.2 cm and 5.8 cm. Select every proposed third-side length that would complete a triangle.

A photo annotation requests a right triangle with a 5 cm hypotenuse and a 6 cm leg. Which check correctly evaluates the request?

A. Two triangles are possible because the leg can face either way.B. No triangle is possible because the hypotenuse must be longest.C. Many triangles are possible because the other leg is not given.D. One triangle is possible because a right angle fixes the shape.

For a shrimp-counting zone, the only conditions are that two sides are each 4.5 cm long. Neither the third side nor any angle is specified. How many different triangles satisfy these conditions?

A. No triangle satisfies the conditions.B. Exactly two triangles satisfy the conditions.C. Exactly one triangle satisfies the conditions.D. Many different triangles satisfy the conditions.

10.

The class also needs a four-sided survey outline. Construct rhombus ABCD with every side 3 cm long and interior angle DAB = 75°. Begin with AB horizontal, with A on the left, and place D above AB. Draw all four sides. Label the vertices, all side lengths, and the 75° angle.

Review the Field Plans

Decide whether each plan gives enough information. Use measurements, not the appearance of a sketch, to justify your decisions.

11.

A rectangular counting window must have a perimeter of 20 cm. No individual side length is specified. How many different rectangles meet the condition?

A. Many different rectangles meet the condition.B. No rectangle meets the condition.C. Exactly two rectangles meet the condition.D. Exactly one rectangle meets the condition.

12.

On a triangular survey sketch, one interior angle is 75°. At another vertex, a side is extended to make an exterior angle of 125°. That exterior angle and the interior angle at the same vertex form a straight angle.

What is the interior angle at the remaining vertex? °

13.

The class needs one fixed size and shape for a triangular photo overlay. Every proposed plan below has interior angles of 28°, 62°, and 90°. Which additional instruction fixes exactly one triangle?

A. Make the side opposite the 28° angle the shortest side.B. Make the side opposite the 90° angle the longest side.C. Make the side opposite the 90° angle exactly 8 cm.D. Make the side opposite the 62° angle longer than 1 cm.

14.

Leo says, “If all our triangular photo outlines have a perimeter of 18 cm, they must have the same size and shape.” Evaluate his claim using two possible sets of side lengths. Explain why both sets form triangles but do not give the same shape.